1. Geometric Sequences

Geometric Sequences

Introduction

A geometric sequence is an ordered list of numbers in which each term is obtained by multiplying the previous term by the same constant value.

For example:

2, 6, 18, 54, 162, ...

Each term is multiplied by 3 to obtain the next term.

2 × 3 = 6

6 × 3 = 18

18 × 3 = 54

54 × 3 = 162

The constant multiplier is called the common ratio.

Geometric sequences appear in many real-world situations, including population growth, compound interest, radioactive decay, repeated discounts, bacterial reproduction, and patterns involving repeated multiplication or division.

Unlike arithmetic sequences, which involve a constant difference, geometric sequences involve a constant ratio.

What Is a Geometric Sequence?

A geometric sequence is a sequence in which the ratio between consecutive terms is constant.

Consider:

3, 9, 27, 81, 243, ...

To move from one term to the next, multiply by 3.

3 × 3 = 9

9 × 3 = 27

27 × 3 = 81

81 × 3 = 243

Therefore, the common ratio is:

r = 3

General Pattern

A geometric sequence can be represented as:

a, ar, ar², ar³, ar⁴, ...

where:

• a is the first term.

• r is the common ratio.

• ar is the second term.

• ar² is the third term.

• ar³ is the fourth term.

For example, if a = 5 and r = 2:

First term: 5

Second term: 5 × 2 = 10

Third term: 5 × 2² = 20

Fourth term: 5 × 2³ = 40

Fifth term: 5 × 2⁴ = 80

The sequence is:

5, 10, 20, 40, 80, ...

Important Observation

A geometric sequence is defined by repeated multiplication, not repeated addition.

The amount added between consecutive terms may change, but the ratio remains constant.

Understanding the Common Ratio

The common ratio is the fixed multiplier used to generate consecutive terms in a geometric sequence.

It is represented by the letter r.

To calculate the common ratio, divide a term by the previous term:

r = a₂/a₁

We can also use:

r = a₃/a₂

or:

r = a₄/a₃

For a geometric sequence with nonzero terms, all these ratios must be equal.

Worked Example

Find the common ratio of:

4, 12, 36, 108, ...

Divide consecutive terms:

12/4 = 3

36/12 = 3

108/36 = 3

All the ratios are equal.

Therefore:

r = 3

Another Example

Find the common ratio of:

5, 20, 80, 320, ...

20/5 = 4

80/20 = 4

320/80 = 4

Therefore:

r = 4

Key Principle

A sequence is geometric when the ratio between consecutive nonzero terms remains constant.

Identifying Geometric Sequences

To determine whether a sequence is geometric, calculate the ratios between consecutive terms.

Worked Example

Determine whether the sequence is geometric:

2, 8, 32, 128, ...

Calculate:

8/2 = 4

32/8 = 4

128/32 = 4

The ratios are constant.

Therefore, the sequence is geometric.

Common ratio: r = 4

Non-Geometric Example

Consider:

2, 5, 8, 11, 14, ...

Calculate:

5/2 = 2.5

8/5 = 1.6

11/8 = 1.375

The ratios are not constant.

Therefore, this sequence is not geometric.

However, its differences are constant:

5 − 2 = 3

8 − 5 = 3

11 − 8 = 3

This is an arithmetic sequence.

Another Non-Geometric Example

Consider:

1, 4, 9, 16, 25, ...

The ratios are:

4/1 = 4

9/4 = 2.25

16/9 ≈ 1.78

The ratios are not constant.

Therefore, the sequence is not geometric.

It is a sequence of square numbers.

Comparing Arithmetic and Geometric Sequences

Arithmetic and geometric sequences both follow predictable patterns, but they use different operations.

Example

Arithmetic sequence:

4, 8, 12, 16, 20, ...

The difference is always 4.

Geometric sequence:

4, 8, 16, 32, 64, ...

The ratio is always 2.

Both begin with 4 and 8, but their later terms are different.

Why Is This Important?

Two sequences may begin with the same values but follow completely different rules.

We must examine the pattern rather than assuming that the next term is obvious.

Generating Terms Using a Common Ratio

Once the first term and common ratio are known, we can generate additional terms.

Worked Example

A geometric sequence has:

a₁ = 6

r = 3

Find the first six terms.

Step 1: Write the First Term

a₁ = 6

Step 2: Multiply by the Common Ratio

a₂ = 6 × 3 = 18

a₃ = 18 × 3 = 54

a₄ = 54 × 3 = 162

a₅ = 162 × 3 = 486

a₆ = 486 × 3 = 1,458

Final Sequence

6, 18, 54, 162, 486, 1,458, ...

Another Example

A geometric sequence has:

a₁ = 100

r = 0.5

Find the first five terms.

a₁ = 100

a₂ = 100 × 0.5 = 50

a₃ = 50 × 0.5 = 25

a₄ = 25 × 0.5 = 12.5

a₅ = 12.5 × 0.5 = 6.25

The sequence is:

100, 50, 25, 12.5, 6.25, ...

This sequence decreases because the common ratio is between 0 and 1.

Geometric Sequences with Ratios Greater Than 1

When the first term is positive and the common ratio is greater than 1, the sequence increases.

Example

2, 4, 8, 16, 32, 64, ...

Common ratio:

r = 2

Each term is twice the previous term.

Growth Pattern

Notice that the differences are:

2, 4, 8, 16, 32

The differences increase rather than remaining constant.

This is characteristic of geometric growth.

Exponential Growth

When a positive quantity is repeatedly multiplied by a factor greater than 1, it experiences exponential growth.

Examples include:

• A population doubling each generation.

• An investment increasing by a fixed percentage.

• A pattern doubling the number of objects at each stage.

• Bacteria reproducing under ideal conditions.

Important Observation

A geometric sequence with r > 1 does not increase by a fixed amount.

It increases by a fixed factor.

Geometric Sequences with Ratios Between 0 and 1

When the first term is positive and the common ratio is between 0 and 1, the sequence decreases.

Example

80, 40, 20, 10, 5, ...

Common ratio:

r = 0.5

Each term is half the previous term.

Another Example

1,000, 800, 640, 512, ...

Calculate the ratio:

800/1,000 = 0.8

640/800 = 0.8

512/640 = 0.8

Therefore:

r = 0.8

Each term is 80% of the previous term.

This means the quantity decreases by 20% each step.

Exponential Decay

Repeated multiplication by a positive ratio less than 1 produces exponential decay.

Examples include:

• Depreciation of equipment.

• Radioactive decay over equal half-life intervals.

• Reduction of a substance by a fixed percentage.

• Decreasing medication concentration in simplified models.

Important Observation

A decreasing geometric sequence with positive terms approaches zero when 0 < r < 1.

The terms become smaller, but they do not become exactly zero after any finite number of steps if the first term is nonzero.

Geometric Sequences with a Negative Common Ratio

A geometric sequence can have a negative common ratio.

When this happens, the signs alternate between positive and negative.

Worked Example

Consider:

3, −6, 12, −24, 48, ...

Calculate the ratios:

−6/3 = −2

12/(−6) = −2

−24/12 = −2

48/(−24) = −2

Therefore:

r = −2

Generating Additional Terms

The next term is:

48 × (−2) = −96

The following term is:

−96 × (−2) = 192

The sequence continues:

3, −6, 12, −24, 48, −96, 192, ...

Why Do the Signs Alternate?

Multiplying a positive number by a negative number produces a negative result.

Multiplying a negative number by a negative number produces a positive result.

Therefore, repeated multiplication by a negative ratio causes alternating signs.

Another Example

Consider:

64, −32, 16, −8, 4, ...

The common ratio is:

r = −0.5

The signs alternate, while the absolute values decrease.

Key Observation

For a nonzero first term and a negative common ratio:

• The signs alternate.

• If |r| > 1, the absolute values increase.

• If 0 < |r| < 1, the absolute values decrease.

• If r = −1, the sequence alternates between two values.

Special Common Ratios

Some common ratios produce interesting patterns.

Common Ratio of 1

Consider:

7, 7, 7, 7, 7, ...

Each term is multiplied by 1.

Therefore:

r = 1

The sequence is constant.

Common Ratio of −1

Consider:

5, −5, 5, −5, 5, ...

Each term is multiplied by −1.

Therefore:

r = −1

The sequence alternates between 5 and −5.

Common Ratio of 0

Consider:

8, 0, 0, 0, 0, ...

Each term is multiplied by 0.

Therefore:

r = 0

This is a geometric sequence under the repeated-multiplication definition, although later ratios such as 0/0 are undefined.

All-Zero Sequence

The sequence:

0, 0, 0, 0, ...

is a special case.

Every term is zero, so it can be generated using any common multiplier. Consequently, the common ratio is not uniquely determined.

In many introductory problems, geometric sequences are restricted to nonzero terms to avoid these complications.

Geometric Sequences with Fractions

Geometric sequences may contain fractions.

Worked Example

Consider:

1/2, 1, 2, 4, 8, ...

Calculate the ratio:

1 ÷ (1/2) = 2

2 ÷ 1 = 2

4 ÷ 2 = 2

Therefore:

r = 2

Another Example

Consider:

3/4, 3/8, 3/16, 3/32, ...

Calculate:

(3/8) ÷ (3/4)

= (3/8) × (4/3)

= 1/2

Therefore:

r = 1/2

Each term is half the previous term.

Generating the Next Terms

The next term is:

3/32 × 1/2 = 3/64

Then:

3/64 × 1/2 = 3/128

The sequence continues:

3/4, 3/8, 3/16, 3/32, 3/64, 3/128, ...

Geometric Sequences with Decimals

Decimal values can also form geometric sequences.

Example

0.2, 0.6, 1.8, 5.4, ...

Calculate:

0.6/0.2 = 3

1.8/0.6 = 3

5.4/1.8 = 3

Therefore:

r = 3

Another Example

10, 7.5, 5.625, 4.21875, ...

Calculate:

7.5/10 = 0.75

5.625/7.5 = 0.75

4.21875/5.625 = 0.75

Therefore:

r = 0.75

Each term is 75% of the previous term.

This corresponds to a 25% decrease each step.

Recursive Rules for Geometric Sequences

A recursive rule describes how to find a term using the previous term.

For a geometric sequence:

aₙ = r × aₙ₋₁, for n ≥ 2

We must also specify the first term.

Worked Example

Consider:

4, 12, 36, 108, ...

The first term is:

a₁ = 4

The common ratio is:

r = 3

The recursive rule is:

a₁ = 4

aₙ = 3aₙ₋₁, for n ≥ 2

Interpreting the Rule

To find the next term, multiply the previous term by 3.

For example:

a₅ = 3 × a₄

a₅ = 3 × 108

a₅ = 324

Another Example

For:

100, 50, 25, 12.5, ...

the recursive rule is:

a₁ = 100

aₙ = 0.5aₙ₋₁, for n ≥ 2

This describes repeated halving.

The nth Term Formula for a Geometric Sequence

Repeated multiplication can also be represented using an explicit formula.

For a geometric sequence:

aₙ = a₁rⁿ⁻¹

where:

• aₙ is the nth term.

• a₁ is the first term.

• r is the common ratio.

• n is the term position.

Why Is the Exponent n − 1?

Consider a geometric sequence beginning with 5 and having common ratio 2.

First term:

a₁ = 5

Second term:

a₂ = 5 × 2

Third term:

a₃ = 5 × 2²

Fourth term:

a₄ = 5 × 2³

The number of multiplications by 2 is always one fewer than the term position.

Therefore:

aₙ = 5 × 2ⁿ⁻¹

Worked Example

Find the sixth term of:

3, 9, 27, 81, ...

Identify:

a₁ = 3

r = 3

n = 6

Substitute:

a₆ = 3 × 3⁵

a₆ = 3 × 243

a₆ = 729

Another Example

Find the eighth term of:

128, 64, 32, 16, ...

a₁ = 128

r = 1/2

n = 8

a₈ = 128 × (1/2)⁷

a₈ = 128 × 1/128

a₈ = 1

Why Is the Formula Useful?

The explicit formula allows us to calculate distant terms without generating every earlier term.

For example, we can calculate the 50th term directly rather than multiplying 49 times.

Recognising Geometric Patterns in Tables

A table of values can help identify geometric sequences.

Consider:

Calculate consecutive ratios:

6/2 = 3

18/6 = 3

54/18 = 3

162/54 = 3

The common ratio is 3.

Therefore, the sequence is geometric.

Another Table

The ratios are:

10/5 = 2

20/10 = 2

35/20 = 1.75

55/35 ≈ 1.57

The ratios are not constant.

Therefore, the sequence is not geometric.

Important Reminder

A sequence must maintain the same ratio throughout the pattern.

A few matching ratios do not prove that every later term follows the same rule.

Graphing Geometric Sequences

Geometric sequences can be represented as points on a coordinate plane.

The horizontal axis represents the term number n.

The vertical axis represents the term value aₙ.

Worked Example

Graph:

2, 4, 8, 16, 32

Create a table:

Plot the coordinates:

(1, 2)

(2, 4)

(3, 8)

(4, 16)

(5, 32)

Interpreting the Graph

Unlike arithmetic sequences, geometric sequences generally do not form straight-line patterns.

Instead, they follow exponential relationships.

For positive initial values and r > 1, the graph rises increasingly rapidly.

For positive initial values and 0 < r < 1, the graph decreases toward zero.

For negative common ratios, the plotted points alternate above and below the horizontal axis.

Discrete Versus Continuous

A geometric sequence is normally represented by discrete points because term numbers are positive integers.

An exponential function can describe a continuous curve, but the sequence includes only the points corresponding to valid term positions.

Real-World Application: Bacterial Growth

Under ideal conditions, some bacteria can reproduce by dividing into two cells.

Suppose a bacterial population begins with 100 cells and doubles every hour.

The population is:

100, 200, 400, 800, 1,600, ...

Identify the Common Ratio

200/100 = 2

400/200 = 2

800/400 = 2

Therefore:

r = 2

Predict the Next Terms

After another doubling:

1,600 × 2 = 3,200

Then:

3,200 × 2 = 6,400

The sequence continues:

100, 200, 400, 800, 1,600, 3,200, 6,400, ...

Interpretation

The population doubles during each equal time interval.

This is geometric growth.

Model Limitation

Real bacterial populations cannot double indefinitely.

Eventually, nutrients, space, waste accumulation, and other environmental factors limit growth.

The geometric model is most appropriate during periods when growth conditions remain approximately constant.

Real-World Application: Compound Interest

Suppose $1,000 is invested in an account that earns 5% interest annually, compounded once per year.

Each year, the balance becomes 105% of the previous balance.

Therefore:

r = 1.05

The account balances at the end of successive years are:

Why Is This Geometric?

Each balance is calculated by multiplying the previous balance by 1.05.

The ratio remains constant.

Formula

If n represents the number of completed years:

Aₙ = 1,000(1.05)ⁿ

Here the initial balance is at n = 0, rather than term position 1.

Important Observation

The interest earned each year increases because the balance increases.

This is different from simple interest, which adds a fixed amount each year.

Real-World Application: Radioactive Decay

Radioactive substances decay over time.

During each half-life, half of the remaining radioactive nuclei are expected to decay.

Suppose a sample initially contains 800 mg of a radioactive substance.

After each half-life, the expected remaining mass is half of the previous amount.

The values are:

800, 400, 200, 100, 50, ...

Common Ratio

400/800 = 0.5

200/400 = 0.5

100/200 = 0.5

Therefore:

r = 0.5

Interpretation

The amount remaining decreases geometrically.

Each term represents the expected remaining amount after another half-life.

Important Scientific Detail

Radioactive decay is random at the level of individual nuclei.

The geometric model describes the expected behaviour of a large population of radioactive nuclei.

Real-World Application: Depreciation

A vehicle loses 20% of its value each year.

Suppose its initial value is $30,000.

Each year, the vehicle retains 80% of its previous value.

Therefore:

r = 0.8

Generate the Values

Initial value:

$30,000

After one year:

30,000 × 0.8 = $24,000

After two years:

24,000 × 0.8 = $19,200

After three years:

19,200 × 0.8 = $15,360

After four years:

15,360 × 0.8 = $12,288

The sequence is:

30,000, 24,000, 19,200, 15,360, 12,288, ...

Interpretation

The vehicle loses 20% of its current value each year.

The dollar amount lost decreases over time because the remaining value becomes smaller.

This is a geometric decay model.

Real-World Application: A Bouncing Ball

Suppose a ball is dropped from a height of 4 metres.

After each bounce, it rises to 75% of its previous maximum height.

The maximum heights after successive bounces are:

4 × 0.75 = 3 m

3 × 0.75 = 2.25 m

2.25 × 0.75 = 1.6875 m

1.6875 × 0.75 = 1.265625 m

The sequence of maximum heights, including the initial height, is:

4, 3, 2.25, 1.6875, 1.265625, ...

Common Ratio

r = 0.75

Interpretation

Each new maximum height is 75% of the previous maximum height.

This is a geometric pattern.

In real situations, the ratio may vary depending on the ball, surface, and conditions.

Describing Geometric Patterns Using Mathematical Language

When describing a geometric sequence, use precise mathematical vocabulary.

Example: Increasing Sequence

3, 12, 48, 192, ...

A suitable description is:

"The sequence is geometric with a first term of 3 and a common ratio of 4. Each term is four times the previous term."

Example: Decreasing Sequence

160, 80, 40, 20, ...

A suitable description is:

"The sequence is geometric with a first term of 160 and a common ratio of 0.5. Each term is half the previous term."

Example: Alternating Sequence

5, −15, 45, −135, ...

A suitable description is:

"The sequence is geometric with a first term of 5 and a common ratio of −3. The signs alternate, and the absolute values triple."

Example: Constant Sequence

9, 9, 9, 9, ...

A suitable description is:

"The sequence is geometric with a common ratio of 1. Every term has the same value."

Useful Mathematical Phrases

• "The common ratio is..."

• "Each term is multiplied by..."

• "The sequence increases geometrically..."

• "The sequence decreases geometrically..."

• "The absolute values increase..."

• "The signs alternate..."

• "The sequence represents exponential growth..."

• "The sequence represents exponential decay..."

• "The ratio between consecutive terms remains constant..."

Common Mistakes

Confusing the common ratio with the common difference

Arithmetic sequences use a constant difference.

Geometric sequences use a constant ratio.

Subtracting instead of dividing

To find the common ratio, divide consecutive terms rather than subtracting them.

Dividing in the wrong order

Use:

r = next term / previous term

For example, in 2, 6, 18, ...:

r = 6/2 = 3

Not 2/6.

Assuming all geometric sequences increase

Geometric sequences may increase, decrease, remain constant, or alternate in sign.

Ignoring negative common ratios

A negative ratio causes alternating signs when the first term is nonzero.

Assuming a common ratio must be a whole number

Ratios may be fractions, decimals, negative numbers, or zero.

Using a constant difference to generate terms

Geometric sequences require multiplication by the common ratio.

Using the wrong exponent

The explicit nth term formula uses n − 1, not n, when a₁ is the first term.

Assuming exponential growth continues forever in real situations

Real-world systems often have limitations that prevent unlimited growth.

Assuming a decreasing geometric sequence reaches zero

For nonzero first terms and 0 < r < 1, the terms approach zero but do not equal zero after any finite number of steps.

Key Terms

• Sequence: An ordered list of numbers following a pattern.

• Term: An individual number in a sequence.

• Term Number: Position of a term in a sequence.

• Geometric Sequence: A sequence generated by multiplying each term by a constant ratio.

• Common Ratio: Constant multiplier between consecutive terms.

• First Term: The value at position n = 1.

• Consecutive Terms: Terms appearing next to each other in a sequence.

• Arithmetic Sequence: Sequence with a constant difference between consecutive terms.

• Common Difference: Fixed amount added to obtain the next term of an arithmetic sequence.

• Recursive Rule: Rule that calculates a term using the previous term.

• Explicit Formula: Formula that calculates a term directly from its position.

• nth Term Formula: Algebraic rule used to calculate the nth term.

• Exponential Growth: Growth involving repeated multiplication by a factor greater than 1.

• Exponential Decay: Decrease involving repeated multiplication by a factor between 0 and 1.

• Alternating Sequence: Sequence whose terms alternate between positive and negative values.

• Constant Sequence: Sequence in which every term has the same value.

• Ratio: Comparison of two quantities through division.

• Multiplier: Number used to multiply another quantity.

• Depreciation: Reduction in value over time.

• Compound Interest: Interest calculated on both the original principal and previously earned interest.

• Half-Life: Time required for half of a radioactive sample's undecayed nuclei to decay on average.

• Discrete Graph: Graph consisting of separate points at specified input values.

• Exponential Function: Function in which the variable appears in an exponent.

• Mathematical Model: Mathematical representation of a real-world situation.

Key Takeaways

• A geometric sequence has a constant ratio between consecutive terms.

• The common ratio is represented by r.

• For nonzero consecutive terms, the common ratio is found by dividing the next term by the previous term.

• A geometric sequence is generated through repeated multiplication.

• Arithmetic sequences use constant differences, while geometric sequences use constant ratios.

• A positive common ratio greater than 1 produces increasing positive terms when the first term is positive.

• A positive common ratio between 0 and 1 produces decreasing positive terms when the first term is positive.

• A negative common ratio produces alternating signs for a nonzero first term.

• A common ratio of 1 produces a constant sequence.

• A common ratio of −1 produces alternating values of equal absolute magnitude.

• The recursive rule is aₙ = raₙ₋₁ for n ≥ 2.

• The explicit formula is aₙ = a₁rⁿ⁻¹.

• Geometric sequences are closely related to exponential functions.

• Geometric sequences can model growth, decay, depreciation, and repeated percentage changes.

• Real-world geometric models depend on assumptions that may not remain valid indefinitely.

Check Your Understanding

Identifying Common Ratios

1. Find the common ratio of 2, 6, 18, 54, ...

2. Find the common ratio of 5, 20, 80, 320, ...

3. Find the common ratio of 100, 50, 25, 12.5, ...

4. Find the common ratio of 3, −9, 27, −81, ...

Identifying Geometric Sequences

5. Is 4, 8, 16, 32, ... geometric?

6. Is 3, 7, 11, 15, ... geometric?

7. Is 81, 27, 9, 3, ... geometric?

8. Is 2, 4, 7, 11, ... geometric?

Generating Terms

9. Find the next three terms of 2, 10, 50, ...

10. Find the next three terms of 64, 32, 16, ...

11. Find the next three terms of 5, −10, 20, ...

12. Generate the first five terms of a geometric sequence with a₁ = 3 and r = 4.

Using Mathematical Rules

13. Write the recursive rule for 2, 6, 18, 54, ...

14. Write the nth term formula for 5, 15, 45, 135, ...

15. Find the sixth term of 2, 4, 8, 16, ...

16. Find the fifth term of 81, 27, 9, 3, ...

Interpreting Geometric Patterns

17. Explain the difference between a common ratio of 2 and a common ratio of 0.5.

18. Describe what happens when the common ratio is negative.

19. A population doubles every hour. What is its common ratio?

20. A product loses 10% of its value each year. What is its common ratio?

Answer Key

Extended Challenge

Identifying Geometric Sequences

1. Determine whether 4, 12, 36, 108, ... is geometric.

2. Determine whether 2, 6, 18, 52, ... is geometric.

3. Determine whether 128, 64, 32, 16, ... is geometric.

4. Determine whether −3, 6, −12, 24, ... is geometric.

5. Determine whether 1, 4, 9, 16, ... is geometric.

Generating Terms

6. Generate the first eight terms of a geometric sequence with a₁ = 2 and r = 3.

7. Generate the first six terms of a geometric sequence with a₁ = 256 and r = 0.5.

8. Generate the first seven terms of a geometric sequence with a₁ = 5 and r = −2.

9. Generate the first six terms of a geometric sequence with a₁ = 1/2 and r = 4.

10. Generate the first six terms of a geometric sequence with a₁ = 1,000 and r = 0.8.

Using nth Term Formulas

11. Find the tenth term of 3, 6, 12, 24, ...

12. Find the eighth term of 5, 15, 45, 135, ...

13. Find the seventh term of 128, 64, 32, 16, ...

14. Find the sixth term of 2, −6, 18, −54, ...

15. Write the nth term formula for 4, 20, 100, 500, ...

Real-World Applications

16. A bacterial population begins with 50 cells and doubles every hour. How many cells are expected after six doublings?

17. An investment begins at $2,000 and increases by 4% annually. Write a formula for its value after n years.

18. A radioactive sample begins with 640 mg and halves every 5 years. How much remains after 20 years?

19. A vehicle begins with a value of $40,000 and loses 15% of its value annually. Write the first five values, including the initial value.

20. A ball rebounds to 80% of its previous maximum height. If dropped from 5 metres, calculate the maximum height after four bounces.

Final Challenge: Investigating a Geometric Sequence

Consider the sequence:

4, 12, 36, 108, 324, ...

Complete the investigation.

1. Identify the first term.

2. Calculate the ratio between the first and second terms.

3. Calculate the ratio between the second and third terms.

4. Explain why the sequence is geometric.

5. Identify the common ratio.

6. Write the recursive rule.

7. Write the explicit nth term formula.

8. Generate the sixth term.

9. Generate the seventh term.

10. Calculate the tenth term.

11. Explain why the differences between consecutive terms are not constant.

12. Compare the sequence with an arithmetic sequence beginning with 4 and having common difference 8.

13. Create a table for the first six terms of both sequences.

14. Plot both sequences on the same coordinate plane.

15. Describe the differences between their graphical patterns.

16. Explain why the geometric sequence grows more rapidly for large term numbers.

17. Determine whether 972 belongs to the geometric sequence.

18. Explain the meaning of the exponent n − 1 in the nth term formula.

19. Describe a real-world situation that could follow this geometric pattern.

20. Identify a limitation of using this model to describe unlimited real-world growth.

Further Investigation: Create Your Own Geometric Sequences

Create three different geometric sequences:

• Sequence A: An increasing sequence with a positive common ratio greater than 1.

• Sequence B: A decreasing positive sequence with a common ratio between 0 and 1.

• Sequence C: An alternating sequence with a negative common ratio.

For each sequence:

1. Identify the first term.

2. Identify the common ratio.

3. Generate the first eight terms.

4. Verify the ratio between consecutive terms.

5. Write the recursive rule.

6. Write the nth term formula.

7. Calculate the tenth term.

8. Describe the behaviour of the sequence.

9. Sketch the graph of its first six terms.

10. Explain how its graph differs from the other two sequences.

2. The nth Term Formula

The nth Term Formula for Geometric Sequences

Introduction

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant value called the common ratio.

Consider the sequence:

3, 6, 12, 24, 48, ...

Each term is multiplied by 2 to obtain the next term.

We can continue this pattern through repeated multiplication. However, what if we want to find the 20th term, the 50th term, or the 100th term?

Calculating every previous term would be time-consuming and unnecessary.

Instead, we can use the nth term formula for geometric sequences to calculate any term directly from its position.

This formula is especially useful for studying exponential growth and decay, compound interest, population changes, depreciation, and repeated percentage changes.

Arithmetic Sequences Versus Geometric Sequences

Before developing the geometric nth term formula, it is important to understand how geometric sequences differ from arithmetic sequences.

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

Example:

4, 7, 10, 13, 16, ...

Each term increases by 3.

The common difference is:

d = 3

The nth term formula is:

aₙ = a₁ + (n − 1)d

Arithmetic sequences involve repeated addition or subtraction.

Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms.

Example:

4, 12, 36, 108, 324, ...

Each term is multiplied by 3.

The common ratio is:

r = 3

Geometric sequences involve repeated multiplication or division.

Comparing the Two Types

Important Observation

Arithmetic sequences change by a constant amount.

Geometric sequences change by a constant factor.

For example:

Arithmetic:

5, 10, 15, 20, 25, ...

Geometric:

5, 10, 20, 40, 80, ...

Both sequences begin with 5 and 10, but they follow different patterns.

Identifying the First Term

The first term of a sequence is represented by a₁.

Consider:

7, 21, 63, 189, ...

The first term is:

a₁ = 7

Another example:

160, 80, 40, 20, ...

The first term is:

a₁ = 160

The first term provides the starting value for the geometric pattern.

Why Is the First Term Important?

Every later term is calculated by multiplying the first term by a power of the common ratio.

Changing the first term changes every value in the sequence.

For example:

Sequence A:

2, 4, 8, 16, ...

Sequence B:

5, 10, 20, 40, ...

Both sequences have a common ratio of 2.

However, their terms differ because their first terms are different.

Identifying the Common Ratio

The common ratio is the constant multiplier between consecutive terms.

It is represented by r.

For consecutive nonzero terms:

r = a₂/a₁

We can also calculate:

r = a₃/a₂

r = a₄/a₃

All ratios must be equal for the sequence to be geometric.

Worked Example

Find the common ratio of:

5, 15, 45, 135, ...

Calculate:

15/5 = 3

45/15 = 3

135/45 = 3

Therefore:

r = 3

Another Example

Find the common ratio of:

96, 48, 24, 12, ...

48/96 = 1/2

24/48 = 1/2

12/24 = 1/2

Therefore:

r = 1/2

Negative Common Ratios

Consider:

4, −8, 16, −32, ...

Calculate:

−8/4 = −2

16/(−8) = −2

−32/16 = −2

Therefore:

r = −2

A negative common ratio causes the signs to alternate when the first term is nonzero.

Developing the nth Term Formula

Consider a geometric sequence with first term a₁ and common ratio r.

The first few terms are:

First term:

a₁

Second term:

a₂ = a₁ × r

Third term:

a₃ = a₁ × r × r = a₁r²

Fourth term:

a₄ = a₁ × r × r × r = a₁r³

Fifth term:

a₅ = a₁r⁴

Notice the pattern:

The exponent is always one less than the term number.

Therefore, the nth term formula is:

aₙ = a₁rⁿ⁻¹

where:

• aₙ = nth term

• a₁ = first term

• r = common ratio

• n = term number

This is called an explicit formula because it allows us to calculate a term directly without finding earlier terms.

Why Does the Formula Use n − 1?

Consider:

2, 6, 18, 54, 162, ...

The first term is 2.

The common ratio is 3.

To reach the second term, multiply by 3 once.

To reach the third term, multiply by 3 twice.

To reach the fourth term, multiply by 3 three times.

To reach the fifth term, multiply by 3 four times.

Therefore:

The exponent counts how many times we multiply by the common ratio.

To reach term n from term 1, we multiply by r exactly n − 1 times.

Important Reminder

Any nonzero number raised to the power zero equals 1.

Therefore:

a₁ = a₁r⁰ = a₁ × 1

This ensures that the formula correctly produces the first term.

Finding the nth Term Formula from a Sequence

Worked Example

Find the nth term formula for:

4, 12, 36, 108, 324, ...

Step 1: Identify the First Term

a₁ = 4

Step 2: Find the Common Ratio

r = 12/4

r = 3

Step 3: Substitute into the Formula

aₙ = a₁rⁿ⁻¹

aₙ = 4 × 3ⁿ⁻¹

Final Formula

aₙ = 4 × 3ⁿ⁻¹

Step 4: Verify

For n = 1:

a₁ = 4 × 3⁰ = 4

For n = 2:

a₂ = 4 × 3¹ = 12

For n = 3:

a₃ = 4 × 3² = 36

The formula correctly describes the sequence.

Another Example: A Decreasing Sequence

Find the nth term formula for:

80, 40, 20, 10, 5, ...

Step 1: Identify the First Term

a₁ = 80

Step 2: Find the Common Ratio

r = 40/80

r = 1/2

Step 3: Write the Formula

aₙ = 80(1/2)ⁿ⁻¹

Step 4: Verify

a₁ = 80(1/2)⁰ = 80

a₂ = 80(1/2)¹ = 40

a₃ = 80(1/2)² = 20

a₄ = 80(1/2)³ = 10

The formula produces the correct terms.

Interpretation

Because the common ratio is between 0 and 1, the terms decrease toward zero.

This is an example of geometric decay.

Finding a Formula with a Negative Common Ratio

Worked Example

Find the nth term formula for:

3, −6, 12, −24, 48, ...

Step 1: Identify the First Term

a₁ = 3

Step 2: Find the Common Ratio

r = −6/3

r = −2

Step 3: Write the Formula

aₙ = 3(−2)ⁿ⁻¹

Step 4: Verify

For n = 1:

a₁ = 3(−2)⁰ = 3

For n = 2:

a₂ = 3(−2)¹ = −6

For n = 3:

a₃ = 3(−2)² = 12

For n = 4:

a₄ = 3(−2)³ = −24

The formula correctly describes the alternating sequence.

Important Observation

The brackets around −2 are essential.

For example:

(−2)² = 4

But:

−2² = −4

These expressions are not equivalent.

Always use brackets when raising a negative common ratio to a power.

Using the nth Term Formula to Calculate Specific Terms

Once the formula is known, substitute the required term number.

Worked Example

Find the eighth term of:

aₙ = 5 × 2ⁿ⁻¹

Substitute n = 8:

a₈ = 5 × 2⁸⁻¹

a₈ = 5 × 2⁷

a₈ = 5 × 128

a₈ = 640

Another Example

Find the sixth term of:

aₙ = 3 × 4ⁿ⁻¹

Substitute n = 6:

a₆ = 3 × 4⁵

a₆ = 3 × 1,024

a₆ = 3,072

Decreasing Sequence Example

Find the seventh term of:

aₙ = 128(1/2)ⁿ⁻¹

a₇ = 128(1/2)⁶

a₇ = 128 × 1/64

a₇ = 2

Predicting Distant Terms

The nth term formula is particularly useful when the required term is far beyond the first few terms.

Worked Example

Consider:

2, 6, 18, 54, ...

Find the 12th term.

Step 1: Identify the First Term and Ratio

a₁ = 2

r = 3

Step 2: Write the Formula

aₙ = 2 × 3ⁿ⁻¹

Step 3: Substitute n = 12

a₁₂ = 2 × 3¹¹

a₁₂ = 2 × 177,147

a₁₂ = 354,294

We have calculated the 12th term without generating the previous 11 terms.

Another Example

Find the 20th term of:

1, 2, 4, 8, 16, ...

a₁ = 1

r = 2

aₙ = 2ⁿ⁻¹

a₂₀ = 2¹⁹

a₂₀ = 524,288

Important Observation

Geometric sequences with a common ratio greater than 1 can grow extremely rapidly.

Even when the first term is small, distant terms may become very large.

Comparing Distant Arithmetic and Geometric Terms

Consider two sequences.

Arithmetic sequence:

2, 4, 6, 8, 10, ...

Geometric sequence:

2, 4, 8, 16, 32, ...

The arithmetic sequence increases by 2 each time.

The geometric sequence doubles each time.

nth Term Formulas

Arithmetic:

aₙ = 2n

Geometric:

bₙ = 2 × 2ⁿ⁻¹ = 2ⁿ

Comparing Values

The geometric sequence eventually becomes much larger.

Why Does This Happen?

Arithmetic growth adds the same amount repeatedly.

Geometric growth multiplies by the same factor repeatedly.

Repeated multiplication by a factor greater than 1 produces exponential growth.

Geometric Sequences with Fractional Ratios

A geometric sequence may have a common ratio that is a fraction.

Worked Example

Find the nth term formula for:

243, 81, 27, 9, 3, ...

First term:

a₁ = 243

Common ratio:

r = 81/243 = 1/3

Therefore:

aₙ = 243(1/3)ⁿ⁻¹

Find the Seventh Term

a₇ = 243(1/3)⁶

a₇ = 243 × 1/729

a₇ = 1/3

Interpretation

Each term is one-third of the previous term.

The sequence decreases rapidly.

Geometric Sequences with Decimal Ratios

Worked Example

Find the nth term formula for:

100, 80, 64, 51.2, ...

First term:

a₁ = 100

Common ratio:

r = 80/100 = 0.8

Therefore:

aₙ = 100(0.8)ⁿ⁻¹

Find the Sixth Term

a₆ = 100(0.8)⁵

a₆ = 100(0.32768)

a₆ = 32.768

Interpretation

Each term is 80% of the previous term.

This corresponds to a 20% decrease per step.

Geometric Sequences with a Common Ratio of 1

Consider:

7, 7, 7, 7, ...

The common ratio is:

r = 1

The nth term formula is:

aₙ = 7 × 1ⁿ⁻¹

Since every power of 1 equals 1:

aₙ = 7

The sequence remains constant.

Special Case: Common Ratio of −1

Consider:

5, −5, 5, −5, ...

The formula is:

aₙ = 5(−1)ⁿ⁻¹

Odd-numbered terms are positive.

Even-numbered terms are negative.

The values alternate between 5 and −5.

Finding a Formula from Two Known Terms

Sometimes we are given two terms but not the first term.

We can still determine the formula.

Worked Example

The third term of a geometric sequence is 20.

The fifth term is 80.

Assume the common ratio is positive.

Find the nth term formula.

Step 1: Write the Known Information

a₃ = 20

a₅ = 80

Step 2: Use the Formula

a₃ = a₁r² = 20

a₅ = a₁r⁴ = 80

Step 3: Divide the Equations

80/20 = r⁴/r²

4 = r²

Because the ratio is positive:

r = 2

Step 4: Find the First Term

a₃ = a₁r²

20 = a₁(2²)

20 = 4a₁

a₁ = 5

Step 5: Write the Formula

aₙ = 5 × 2ⁿ⁻¹

Verification

a₃ = 5 × 2² = 20

a₅ = 5 × 2⁴ = 80

The formula is correct.

Important Observation

If the common ratio had not been specified as positive, r = −2 would also satisfy the two given terms.

Two nonconsecutive terms may not always determine a unique geometric sequence.

Finding the Term Number from a Given Value

We can also use the nth term formula to determine which term has a particular value.

Worked Example

Consider:

aₙ = 3 × 2ⁿ⁻¹

Which term has a value of 96?

Step 1: Substitute the Value

96 = 3 × 2ⁿ⁻¹

Step 2: Divide by 3

32 = 2ⁿ⁻¹

Step 3: Express 32 as a Power of 2

32 = 2⁵

Therefore:

n − 1 = 5

n = 6

The sixth term is 96.

Verification

a₆ = 3 × 2⁵

a₆ = 3 × 32

a₆ = 96

Another Example

Which term of:

aₙ = 160(1/2)ⁿ⁻¹

has a value of 10?

10 = 160(1/2)ⁿ⁻¹

1/16 = (1/2)ⁿ⁻¹

Since:

1/16 = (1/2)⁴

n − 1 = 4

n = 5

The fifth term is 10.

Real-World Application: Compound Interest

Suppose $2,000 is invested at an annual interest rate of 5%, compounded once per year.

Each year, the investment is multiplied by:

1 + 0.05 = 1.05

Therefore:

r = 1.05

Writing the Formula

If n represents the number of completed years, starting at n = 0:

Aₙ = 2,000(1.05)ⁿ

This formula uses n rather than n − 1 because the initial investment is recorded at year 0.

Calculate the Value After 10 Years

A₁₀ = 2,000(1.05)¹⁰

A₁₀ ≈ 2,000(1.6288946)

A₁₀ ≈ $3,257.79

Interpretation

The investment grows because each year's interest is calculated using the updated balance.

The model assumes the interest rate remains constant and no money is added or withdrawn.

Real-World Application: Bacterial Growth

A bacterial population begins with 500 cells.

Under ideal conditions, the population doubles every hour.

The common ratio is:

r = 2

Writing the Formula

Let Pₙ represent the expected population after n completed hours.

Pₙ = 500 × 2ⁿ

The exponent is n because the starting population occurs at hour 0.

Predict the Population After 8 Hours

P₈ = 500 × 2⁸

P₈ = 500 × 256

P₈ = 128,000 cells

Interpretation

The population increases exponentially.

However, this model assumes unlimited resources and constant growth conditions.

Real populations eventually experience environmental limitations.

Real-World Application: Radioactive Decay

A radioactive sample initially contains 640 mg of a radioactive isotope.

Its half-life is 5 years.

After every 5-year interval, the expected remaining mass is multiplied by 0.5.

Writing the Formula

Let n represent the number of completed half-lives.

Mₙ = 640(0.5)ⁿ

Find the Expected Mass After 20 Years

Number of half-lives:

n = 20/5 = 4

Substitute:

M₄ = 640(0.5)⁴

M₄ = 640(1/16)

M₄ = 40 mg

Interpretation

After 20 years, approximately 40 mg remains according to the model.

This is an expected value for a sufficiently large sample, since individual radioactive decays are random.

Real-World Application: Depreciation

A vehicle is purchased for $30,000.

Its value decreases by 15% each year.

Each year, it retains:

100% − 15% = 85%

Therefore:

r = 0.85

Writing the Formula

Let Vₙ represent the vehicle's value after n years.

Vₙ = 30,000(0.85)ⁿ

Find the Value After 5 Years

V₅ = 30,000(0.85)⁵

V₅ ≈ 30,000(0.4437053)

V₅ ≈ $13,311.16

Interpretation

The vehicle loses 15% of its remaining value each year.

The amount lost in dollars decreases as the vehicle's value decreases.

This is geometric decay.

A Complete Strategy for nth Term Problems

When solving a geometric sequence problem:

1. Identify the sequence type. Check whether the ratio between consecutive terms is constant.

2. Find the first term. Record a₁.

3. Find the common ratio. Divide a term by the previous nonzero term.

4. Write the nth term formula. Use aₙ = a₁rⁿ⁻¹.

5. Substitute the required term number. Replace n with the desired position.

6. Calculate the exponent first. Follow the order of operations.

7. Multiply by the first term. Complete the calculation.

8. Check the sign. Negative common ratios may produce alternating signs.

9. Verify when possible. Calculate known terms to confirm the formula.

10. Interpret the result. Include appropriate units in real-world problems.

Common Mistakes

Using the arithmetic nth term formula

Geometric sequences require multiplication by a common ratio, not repeated addition.

Subtracting to find the common ratio

Divide consecutive terms rather than subtracting them.

Using the wrong ratio

Calculate:

r = next term / previous term

Forgetting the exponent n − 1

The first term corresponds to zero multiplications by the common ratio.

Confusing the term number with the exponent

For the tenth term, the exponent is 9 when using aₙ = a₁rⁿ⁻¹.

Ignoring brackets around negative ratios

(−2)² = 4, but −2² = −4.

Assuming all geometric sequences increase

Sequences may decrease, remain constant, or alternate in sign.

Confusing percentage decreases with the common ratio

A 20% decrease corresponds to r = 0.8, not r = 0.2.

Confusing year-zero models with term-one models

A quantity recorded at time zero commonly uses Aₙ = A₀rⁿ.

A sequence beginning at term 1 uses aₙ = a₁rⁿ⁻¹.

Failing to verify the formula

Always check that the formula reproduces the known terms.

Key Terms

• Arithmetic Sequence: Sequence with a constant difference between consecutive terms.

• Geometric Sequence: Sequence with a constant ratio between consecutive terms.

• Common Difference: Fixed amount added to generate the next arithmetic term.

• Common Ratio: Fixed multiplier used to generate the next geometric term.

• First Term: Value at position n = 1.

• nth Term: Value at position n in a sequence.

• Term Number: Position of a term.

• Explicit Formula: Formula that calculates a term directly from its position.

• Recursive Formula: Rule that calculates a term using the previous term.

• Exponent: Number indicating repeated multiplication.

• Base: Number being raised to a power.

• Exponential Growth: Growth caused by repeated multiplication by a factor greater than 1.

• Exponential Decay: Decrease caused by repeated multiplication by a factor between 0 and 1.

• Alternating Sequence: Sequence whose terms alternate in sign.

• Constant Sequence: Sequence in which every term has the same value.

• Compound Interest: Interest calculated on both the original investment and previously earned interest.

• Depreciation: Reduction in value over time.

• Half-Life: Time required for half of a radioactive sample's undecayed nuclei to decay on average.

• Prediction: Calculation of an unknown value using a mathematical rule.

• Verification: Checking that a formula produces the expected terms.

• Mathematical Model: Mathematical representation of a real-world situation.

Key Takeaways

• Arithmetic sequences use constant differences.

• Geometric sequences use constant ratios.

• The common ratio is calculated by dividing consecutive nonzero terms.

• The first term is represented by a₁.

• The common ratio is represented by r.

• The geometric nth term formula is aₙ = a₁rⁿ⁻¹.

• The exponent n − 1 counts the number of multiplications from the first term.

• A common ratio greater than 1 produces increasing positive terms when the first term is positive.

• A positive ratio between 0 and 1 produces decreasing positive terms.

• A negative ratio produces alternating signs for a nonzero first term.

• The nth term formula allows distant terms to be calculated directly.

• Geometric sequences are closely related to exponential functions.

• Compound interest, population growth, depreciation, and radioactive decay can be modelled using geometric relationships.

• Real-world formulas must be interpreted according to their starting position and assumptions.

Check Your Understanding

Arithmetic Versus Geometric

1. Explain the difference between arithmetic and geometric sequences.

2. Is 4, 8, 12, 16, ... arithmetic or geometric?

3. Is 4, 8, 16, 32, ... arithmetic or geometric?

4. Find the common ratio of 5, 20, 80, 320, ...

Finding nth Term Formulas

5. Find the nth term formula for 2, 6, 18, 54, ...

6. Find the nth term formula for 5, 10, 20, 40, ...

7. Find the nth term formula for 81, 27, 9, 3, ...

8. Find the nth term formula for 3, −9, 27, −81, ...

Calculating Specific Terms

9. Find the sixth term of aₙ = 2 × 3ⁿ⁻¹.

10. Find the eighth term of aₙ = 5 × 2ⁿ⁻¹.

11. Find the fifth term of aₙ = 160(1/2)ⁿ⁻¹.

12. Find the sixth term of aₙ = 4(−2)ⁿ⁻¹.

Predicting Distant Terms

13. Find the tenth term of 3, 6, 12, 24, ...

14. Find the twelfth term of 1, 3, 9, 27, ...

15. Find the eighth term of 256, 128, 64, 32, ...

16. Find the twentieth term of 2, 4, 8, 16, ...

Applications

17. A population begins with 100 cells and doubles every hour. How many cells are expected after six doublings?

18. An investment of $1,000 grows by 10% annually. Write a formula for its value after n years.

19. A vehicle worth $20,000 loses 20% of its value annually. Find its value after three years.

20. Explain why geometric sequences can grow much faster than arithmetic sequences.

Answer Key

Extended Challenge

Finding Formulas

1. Find the nth term formula for 7, 21, 63, 189, ...

2. Find the nth term formula for 96, 48, 24, 12, ...

3. Find the nth term formula for 5, −15, 45, −135, ...

4. Find the nth term formula for 1/2, 1, 2, 4, ...

5. Find the nth term formula for 100, 75, 56.25, 42.1875, ...

Calculating Distant Terms

6. Find the 15th term of 2, 4, 8, 16, ...

7. Find the 12th term of 5, 15, 45, 135, ...

8. Find the 10th term of 1, −2, 4, −8, ...

9. Find the 20th term of 3, 9, 27, 81, ...

10. Find the 12th term of 4, 2, 1, 0.5, ...

Finding Unknown Values

11. The third term is 24 and the fifth term is 216. If the common ratio is positive, find the nth term formula.

12. The second term is 10 and the fourth term is 90. If the common ratio is positive, find the first term.

13. Which term of 3, 6, 12, 24, ... has value 768?

14. Which term of 256, 128, 64, 32, ... has value 4?

15. Determine whether 486 belongs to the sequence 2, 6, 18, 54, ...

Real-World Applications

16. A bacterial population begins with 200 cells and triples every hour. Find the expected population after five hours.

17. An investment of $5,000 grows by 6% annually. Find its value after 10 years.

18. A vehicle worth $40,000 loses 12% of its value each year. Find its value after five years.

19. A radioactive sample begins with 1,024 mg and halves every four years. Find the expected mass after 20 years.

20. A ball rebounds to 70% of its previous maximum height. If dropped from 10 metres, calculate its maximum height after five bounces.

Final Challenge: Comparing Arithmetic and Geometric Growth

Two sequences begin with the same first term.

Sequence A: 5, 10, 15, 20, 25, ...

Sequence B: 5, 10, 20, 40, 80, ...

Complete the investigation.

1. Identify the first term of each sequence.

2. Determine the common difference of Sequence A.

3. Determine the common ratio of Sequence B.

4. Write the nth term formula for Sequence A.

5. Write the nth term formula for Sequence B.

6. Generate the first eight terms of both sequences.

7. Calculate the tenth term of each sequence.

8. Calculate the fifteenth term of each sequence.

9. Calculate the twentieth term of each sequence.

10. Compare the growth of the two sequences.

11. Explain why the sequences begin with the same first two terms.

12. Determine the first term position at which Sequence B becomes greater than Sequence A.

13. Create a table comparing the first ten terms.

14. Plot the first eight terms of both sequences.

15. Describe the graphical pattern of Sequence A.

16. Describe the graphical pattern of Sequence B.

17. Explain why the geometric sequence eventually grows much faster.

18. Describe a real-world situation that could be modelled by Sequence A.

19. Describe a real-world situation that could be modelled by Sequence B.

20. Explain why understanding the difference between linear and exponential growth is important.

Further Investigation: Create Your Own Geometric Model

Develop a realistic situation involving geometric growth or decay.

Possible contexts include:

• Compound interest.

• Population growth.

• Depreciation.

• Radioactive decay.

• Repeated percentage discounts.

• Bouncing balls.

Then:

1. Describe the situation.

2. Identify the initial value.

3. Determine the common ratio.

4. Explain whether the pattern represents growth or decay.

5. Write the first six terms.

6. Write the nth term formula using an appropriate starting index.

7. Calculate the value after 10 time intervals.

8. Calculate the value after 20 time intervals.

9. Compare the geometric model with an arithmetic model.

10. Explain which model is more appropriate and why.

11. Identify assumptions in the model.

12. Explain when the model might no longer be realistic.

3. Graphing Geometric Sequences

Graphing Geometric Sequences

Introduction

A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant value called the common ratio.

For example:

2, 4, 8, 16, 32, ...

Each term is multiplied by 2.

We can represent this sequence in several ways:

• As an ordered list of numbers.

• As a table of values.

• As an nth term formula.

• As points on a coordinate plane.

Graphing geometric sequences helps us understand how quantities change through repeated multiplication.

Unlike arithmetic sequences, which generally produce straight-line patterns, geometric sequences are associated with exponential relationships.

These relationships are important in mathematics, finance, biology, physics, and many other fields.

Reviewing Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms.

Consider:

3, 9, 27, 81, 243, ...

Find the ratio between consecutive terms:

9/3 = 3

27/9 = 3

81/27 = 3

243/81 = 3

The common ratio is:

r = 3

The nth term formula for a geometric sequence is:

aₙ = a₁rⁿ⁻¹

where:

• aₙ represents the nth term.

• a₁ represents the first term.

• r represents the common ratio.

• n represents the term position.

Example

For:

3, 9, 27, 81, ...

a₁ = 3

r = 3

Therefore:

aₙ = 3 × 3ⁿ⁻¹

This formula allows us to calculate the value at any positive integer term position.

Creating a Table of Values

A table of values organises the term numbers and their corresponding values.

Consider:

2, 6, 18, 54, 162, ...

The common ratio is:

r = 3

Creating the Table

The first column shows the position of each term.

The second column shows its value.

Why Are Tables Useful?

Tables allow us to:

• Organise sequence information.

• Identify the common ratio.

• Generate ordered pairs for graphing.

• Compare values at different positions.

• Recognise patterns of growth or decay.

• Verify the nth term formula.

Extending the Table

Continue multiplying by 3:

The values increase rapidly because each term is three times the previous term.

Creating Tables from nth Term Formulas

Sometimes we are given a formula instead of a list of terms.

We can substitute different term numbers into the formula to create a table.

Worked Example

Create a table for the first five terms of:

aₙ = 4 × 2ⁿ⁻¹

Step 1: Substitute n = 1

a₁ = 4 × 2⁰

a₁ = 4

Step 2: Substitute n = 2

a₂ = 4 × 2¹

a₂ = 8

Step 3: Substitute n = 3

a₃ = 4 × 2²

a₃ = 16

Step 4: Substitute n = 4

a₄ = 4 × 2³

a₄ = 32

Step 5: Substitute n = 5

a₅ = 4 × 2⁴

a₅ = 64

Completed Table

The table is now ready to be used for graphing.

Turning a Table into Coordinates

Each row of a table can be written as an ordered pair.

An ordered pair has the form:

(x, y)

For geometric sequences:

• x represents the term number n.

• y represents the term value aₙ.

Consider:

4, 8, 16, 32, 64

The coordinates are:

(1, 4)

(2, 8)

(3, 16)

(4, 32)

(5, 64)

Important Observation

The first coordinate represents the term position.

The second coordinate represents the value at that position.

For example:

(3, 16)

means that the third term has a value of 16.

Graphing a Geometric Sequence

To graph a geometric sequence, plot its ordered pairs on a coordinate plane.

Worked Example

Graph:

2, 4, 8, 16, 32

Step 1: Create a Table

Step 2: Write the Coordinates

(1, 2)

(2, 4)

(3, 8)

(4, 16)

(5, 32)

Step 3: Label the Axes

Horizontal axis: term number n.

Vertical axis: term value aₙ.

Step 4: Plot the Points

Step 5: Interpret the Graph

Notice that:

• The points rise from left to right.

• The vertical distance between consecutive points increases.

• Each term is twice the previous term.

• The points follow an exponential growth pattern.

The graph illustrates repeated multiplication rather than repeated addition.

Why Geometric Sequences Produce Exponential Patterns

Consider:

3, 6, 12, 24, 48, ...

The common ratio is 2.

Each time the term number increases by 1, the value doubles.

Comparing Changes

The differences increase.

However, the ratio remains constant.

What Does This Mean for the Graph?

The graph does not rise by the same vertical amount each time.

Instead, the vertical changes become larger as the terms increase.

This produces the increasingly steep pattern associated with exponential growth.

Important Distinction

Arithmetic sequences have constant first differences.

Geometric sequences have constant ratios.

For nonconstant positive geometric sequences, the points do not lie on a straight line.

Exponential Growth

A geometric sequence exhibits exponential growth when its positive terms increase through repeated multiplication by a factor greater than 1.

Example

5, 10, 20, 40, 80, 160, ...

First term:

a₁ = 5

Common ratio:

r = 2

nth term formula:

aₙ = 5 × 2ⁿ⁻¹

Table of Values

Interpreting the Graph

• The points rise increasingly rapidly.

• The values double with each new term.

• The differences between consecutive values become larger.

• The pattern represents exponential growth.

Common Ratios Greater Than 1

For a positive first term:

• r = 1.1 means a 10% increase per term.

• r = 1.5 means a 50% increase per term.

• r = 2 means the value doubles.

• r = 3 means the value triples.

Larger common ratios generally produce faster growth when the first term is the same.

Exponential Decay

A geometric sequence exhibits exponential decay when its positive terms decrease through repeated multiplication by a factor between 0 and 1.

Example

160, 80, 40, 20, 10, 5, ...

First term:

a₁ = 160

Common ratio:

r = 0.5

nth term formula:

aₙ = 160(0.5)ⁿ⁻¹

Table of Values

Graph Behaviour

• The points fall from left to right.

• The decreases become smaller in absolute size.

• The values approach zero.

• The sequence remains positive.

Important Observation

For a positive first term and 0 < r < 1, the terms approach zero as n increases.

However, they do not become exactly zero after any finite number of steps.

Comparing Different Rates of Exponential Decay

Consider two geometric sequences.

Sequence A:

100, 80, 64, 51.2, ...

Common ratio:

r = 0.8

Sequence B:

100, 50, 25, 12.5, ...

Common ratio:

r = 0.5

Comparison Table

Interpretation

Both sequences decrease.

However, Sequence B decreases faster because its common ratio is smaller.

Sequence A retains 80% of its previous value.

Sequence B retains only 50%.

Important Conclusion

For positive sequences with the same first term, a smaller positive common ratio between 0 and 1 produces faster decay.

Geometric Sequences with Negative Common Ratios

A geometric sequence may have a negative common ratio.

Consider:

2, −4, 8, −16, 32, ...

The common ratio is:

r = −2

Table of Values

Graph Behaviour

The plotted points alternate above and below the horizontal axis.

The absolute values increase because:

|r| = 2 > 1

Another Example

64, −32, 16, −8, 4, ...

Common ratio:

r = −0.5

The signs alternate, but the absolute values decrease.

Important Observation

A negative common ratio does not produce ordinary monotonic exponential growth or decay.

Instead, the sequence alternates in sign.

The absolute values may grow, decrease, or remain constant depending on |r|.

Geometric Sequences with a Common Ratio of 1

Consider:

7, 7, 7, 7, 7, ...

Common ratio:

r = 1

The nth term formula is:

aₙ = 7

Graph Behaviour

All points have the same vertical coordinate.

The graph forms a horizontal pattern.

This is a constant geometric sequence.

Special Case

A constant nonzero sequence is both:

• Arithmetic, with common difference d = 0.

• Geometric, with common ratio r = 1.

This is an important exception to the usual distinction between linear and exponential patterns.

Discrete and Continuous Graphs

Geometric sequences are normally represented by discrete points.

This is because term numbers are positive integers:

n = 1, 2, 3, 4, ...

Discrete Sequence

Consider:

aₙ = 2ⁿ

The first four terms are:

2, 4, 8, 16

The graph contains:

(1, 2), (2, 4), (3, 8), (4, 16)

There is no term at n = 2.5.

Continuous Exponential Function

Consider:

y = 2ˣ

This function is defined for all real x-values.

Its graph is a continuous exponential curve.

For example, at x = 2.5:

y = 2²⋅⁵ ≈ 5.657

This point belongs to the continuous exponential function but not to the discrete sequence indexed by positive integers.

Why Shouldn't We Connect the Points?

Connecting geometric sequence points with a solid curve can incorrectly suggest that fractional term positions are included.

A smooth curve may be drawn as a visual guide to the underlying exponential relationship, but the sequence itself consists of separate points.

Comparing Arithmetic and Geometric Graphs

Arithmetic and geometric sequences often behave very differently when graphed.

Consider two sequences.

Arithmetic:

2, 4, 6, 8, 10, 12, ...

Geometric:

2, 4, 8, 16, 32, 64, ...

Comparison Table

Comparing the Graphs

Arithmetic Graph

• The points lie on a straight line.

• The vertical change is constant.

• The common difference determines the slope.

• The pattern represents linear growth.

Geometric Graph

• The points follow an exponential pattern.

• The vertical change is not constant.

• The common ratio determines the multiplicative change.

• With a positive first term and r > 1, the values grow increasingly rapidly.

Important Conclusion

Linear growth adds a fixed amount.

Exponential growth multiplies by a fixed factor.

For positive starting values and a fixed growth ratio greater than 1, geometric growth eventually exceeds arithmetic growth with a fixed positive common difference.

Comparing Arithmetic and Geometric Growth Over Time

Suppose two savings plans begin with $100.

Plan A: Arithmetic Growth

The balance increases by $50 each year.

Sequence:

100, 150, 200, 250, 300, ...

Formula:

aₙ = 100 + 50(n − 1)

Plan B: Geometric Growth

The balance increases by 50% each year.

Sequence:

100, 150, 225, 337.50, 506.25, ...

Formula:

bₙ = 100(1.5)ⁿ⁻¹

Comparison

Interpretation

The arithmetic plan increases by the same dollar amount each year.

The geometric plan increases by the same percentage each year.

As the balance grows, the percentage increase produces larger dollar increases.

This causes the geometric sequence to grow more rapidly.

Interpreting Geometric Graphs in Real-World Situations

When interpreting a geometric sequence graph, consider:

• What does the horizontal axis represent?

• What does the vertical axis represent?

• What is the initial value?

• What is the common ratio?

• Does the quantity grow or decay?

• What does each plotted point represent?

• Are the values realistic?

• What assumptions does the model make?

These questions help connect mathematical patterns to meaningful real-world situations.

Real-World Application: Bacterial Growth

A bacterial population begins with 100 cells and doubles every hour under ideal conditions.

Create a Table

Formula

Let P(t) represent the expected population after t hours.

P(t) = 100 × 2ᵗ

Here the initial population is recorded at t = 0.

Interpreting the Graph

• Horizontal axis: time in hours.

• Vertical axis: bacterial population.

• Initial value: 100 cells.

• Common ratio: 2 per hour.

• Growth pattern: exponential.

Prediction

After 8 hours:

P(8) = 100 × 2⁸

P(8) = 100 × 256

P(8) = 25,600 cells

Model Limitation

Real bacterial populations do not grow exponentially forever.

Nutrient availability, space, waste accumulation, and other environmental factors eventually limit growth.

Real-World Application: Compound Interest

An investment begins with $2,000 and earns 5% interest annually, compounded once per year.

Each year, the balance is multiplied by:

1.05

Table of Values

Formula

A(t) = 2,000(1.05)ᵗ

Interpreting the Graph

• Horizontal axis: years elapsed.

• Vertical axis: investment value.

• Initial value: $2,000.

• Common ratio: 1.05.

• Percentage growth: 5% annually.

Prediction

After 10 years:

A(10) = 2,000(1.05)¹⁰

A(10) ≈ $3,257.79

Important Observation

The dollar amount of interest earned increases over time because interest is calculated on the growing balance.

Real-World Application: Radioactive Decay

A radioactive sample initially contains 800 mg of a radioactive isotope.

Its half-life is 5 years.

After every 5-year interval, the expected remaining mass is multiplied by 0.5.

Create a Table

Formula

Let M(t) represent the expected remaining mass after t years.

M(t) = 800(0.5)ᵗ⁄⁵

Interpreting the Graph

• Horizontal axis: time in years.

• Vertical axis: expected remaining mass.

• Initial value: 800 mg.

• Common ratio: 0.5 every five years.

• Pattern: exponential decay.

Prediction

After 30 years:

M(30) = 800(0.5)⁶

M(30) = 800/64

M(30) = 12.5 mg

Scientific Interpretation

Radioactive decay is random at the level of individual nuclei.

The exponential model describes the expected behaviour of a sufficiently large sample.

Real-World Application: Vehicle Depreciation

A vehicle is purchased for $30,000.

Its value decreases by 20% each year.

Each year, the vehicle retains:

100% − 20% = 80%

Therefore:

r = 0.8

Table of Values

Formula

V(t) = 30,000(0.8)ᵗ

Interpreting the Graph

• The graph decreases from left to right.

• The value decreases by 20% annually.

• The dollar amount lost becomes smaller each year.

• The value approaches zero in the mathematical model.

Model Limitation

Real vehicle values may not follow a constant depreciation rate.

Market conditions, vehicle condition, age, and demand affect actual prices.

Real-World Application: A Bouncing Ball

A ball is dropped from a height of 5 metres.

After each bounce, it rises to 70% of its previous maximum height.

Table of Values

Formula

Let H represent the maximum height after n bounces.

H = 5(0.7)ⁿ

Interpreting the Graph

• Horizontal axis: bounce number.

• Vertical axis: maximum height.

• Initial height: 5 m.

• Common ratio: 0.7.

• Pattern: exponential decay.

Important Observation

The heights decrease geometrically because each bounce reaches a fixed percentage of the previous height.

The graph is discrete because bounce numbers are whole numbers.

Finding the Common Ratio from a Graph

We can determine the common ratio by reading consecutive term values from a graph.

Worked Example

A graph contains the points:

(1, 4)

(2, 12)

(3, 36)

(4, 108)

Find the common ratio.

Step 1: Identify Consecutive Terms

a₁ = 4

a₂ = 12

Step 2: Divide

r = 12/4

r = 3

Step 3: Verify

36/12 = 3

108/36 = 3

The ratios are constant.

Therefore, the sequence is geometric.

nth Term Formula

aₙ = 4 × 3ⁿ⁻¹

Finding the Common Ratio from Nonconsecutive Points

Sometimes the graph provides two points that are not consecutive.

Worked Example

A geometric sequence graph contains:

(2, 6) and (5, 162)

Assume the common ratio is positive.

Step 1: Identify the Terms

a₂ = 6

a₅ = 162

Step 2: Compare the Terms

From term 2 to term 5, there are three multiplications by r.

Therefore:

162 = 6r³

Step 3: Solve

r³ = 162/6

r³ = 27

r = 3

Step 4: Find the First Term

a₂ = a₁r

6 = 3a₁

a₁ = 2

Final Formula

aₙ = 2 × 3ⁿ⁻¹

Verification

a₅ = 2 × 3⁴

a₅ = 2 × 81

a₅ = 162

The formula matches the graph.

Recognising Geometric Sequences from Graphs

A geometric sequence graph may show:

• Rapidly increasing positive values.

• Positive values decreasing toward zero.

• Alternating positive and negative values.

• A constant horizontal pattern.

However, appearance alone is not sufficient to prove that a sequence is geometric.

How to Verify

1. Read the term values from the graph.

2. Identify consecutive nonzero terms.

3. Divide each term by the previous term.

4. Check whether the ratios are constant.

Example

A graph contains:

(1, 3), (2, 9), (3, 27), (4, 81)

The ratios are:

9/3 = 3

27/9 = 3

81/27 = 3

Therefore, the sequence is geometric.

Non-Geometric Example

A graph contains:

(1, 2), (2, 5), (3, 10), (4, 17)

The ratios are:

5/2 = 2.5

10/5 = 2

17/10 = 1.7

The ratios are not constant.

Therefore, the sequence is not geometric.

Choosing an Appropriate Graph Scale

Geometric sequences can grow very quickly.

Consider:

2, 6, 18, 54, 162, 486, ...

The sixth term is much larger than the first.

If the vertical axis only extends to 100, the later terms will not fit.

Strategies

• Choose a vertical scale that includes all required values.

• Use intervals such as 10, 50, 100, or 500 where appropriate.

• Label axes clearly.

• Avoid changing the scale partway through an axis.

• Consider graphing fewer terms if the values become extremely large.

• Use logarithmic scales for advanced comparisons across very large ranges.

Important Observation

A poorly chosen scale can make an exponential pattern difficult to interpret.

Always inspect the range of values before drawing the graph.

Common Mistakes

Switching the axes

The term number belongs on the horizontal axis.

The term value belongs on the vertical axis.

Connecting sequence points with a continuous curve

A geometric sequence is normally discrete.

Confusing the common ratio with the common difference

Geometric sequences use repeated multiplication, not repeated addition.

Assuming all geometric graphs increase

Positive geometric sequences may increase, decrease, or remain constant.

Ignoring negative common ratios

Negative ratios produce alternating signs for nonzero terms.

Using an inappropriate graph scale

Exponential growth can produce very large values after only a few terms.

Confusing exponential decay with linear decrease

Exponential decay involves repeated multiplication by a factor between 0 and 1.

Assuming an exponential decay sequence reaches zero

Positive geometric sequences with 0 < r < 1 approach zero but do not become exactly zero after finitely many terms.

Using the wrong starting index

Some real-world models begin at time zero.

Standard sequence notation often begins at n = 1.

Interpreting a graph beyond realistic conditions

Real-world growth and decay models depend on assumptions that may eventually fail.

Key Terms

• Geometric Sequence: Sequence with a constant ratio between consecutive terms.

• Common Ratio: Fixed multiplier used to generate the next term.

• First Term: Value at position n = 1.

• nth Term: Value at position n.

• Table of Values: Organised list of input and output values.

• Ordered Pair: Pair of coordinates written as (x, y).

• Coordinate Plane: Plane formed by horizontal and vertical axes.

• Horizontal Axis: Axis representing the term number or independent variable.

• Vertical Axis: Axis representing the term value or dependent variable.

• Exponential Growth: Increase caused by repeated multiplication by a factor greater than 1.

• Exponential Decay: Decrease caused by repeated multiplication by a factor between 0 and 1.

• Discrete Graph: Graph containing separate points at specified input values.

• Continuous Graph: Graph defined across an interval of real values.

• Arithmetic Sequence: Sequence with a constant difference between consecutive terms.

• Linear Relationship: Relationship with a constant rate of additive change.

• Exponential Relationship: Relationship involving repeated multiplication.

• Initial Value: Starting value of a model.

• Growth Factor: Multiplier greater than 1 representing repeated growth.

• Decay Factor: Multiplier between 0 and 1 representing repeated decay.

• Depreciation: Decrease in value over time.

• Compound Interest: Interest calculated on both the original investment and previously earned interest.

• Half-Life: Time required for half of a radioactive sample's undecayed nuclei to decay on average.

• Mathematical Model: Mathematical representation of a real-world situation.

• Prediction: Calculation of an unknown value using a mathematical relationship.

• Graph Scale: Numerical intervals used along graph axes.

Key Takeaways

• Geometric sequences can be represented using tables, formulas, and graphs.

• The term number is plotted on the horizontal axis.

• The term value is plotted on the vertical axis.

• Geometric sequences have a constant ratio between consecutive terms.

• The nth term formula is aₙ = a₁rⁿ⁻¹.

• Geometric sequence graphs generally follow exponential patterns.

• Positive common ratios greater than 1 produce exponential growth for positive first terms.

• Positive common ratios between 0 and 1 produce exponential decay for positive first terms.

• Negative common ratios produce alternating signs.

• Arithmetic sequence graphs form linear patterns.

• Geometric sequence graphs are normally discrete.

• Geometric growth can eventually exceed linear growth by a large amount.

• The common ratio can be calculated from graph coordinates.

• Tables help organise values before graphing.

• Real-world geometric graphs can model populations, investments, depreciation, and radioactive decay.

• Mathematical models must be interpreted within realistic conditions.

Check Your Understanding

Tables of Values

1. Create a table for the first five terms of 2, 4, 8, 16, ...

2. Create a table for the first five terms of aₙ = 3 × 2ⁿ⁻¹.

3. Create a table for the first five terms of aₙ = 80(0.5)ⁿ⁻¹.

4. Write the ordered pairs for 5, 15, 45, 135, ...

Graphing

5. Which variable belongs on the horizontal axis?

6. Which variable belongs on the vertical axis?

7. Why are geometric sequence graphs normally discrete?

8. Describe the graph of 4, 8, 16, 32, ...

Growth and Decay

9. Does a common ratio of 3 represent growth or decay for a positive first term?

10. Does a common ratio of 0.25 represent growth or decay for a positive first term?

11. Describe the graph of 100, 50, 25, 12.5, ...

12. Describe the graph of 3, −6, 12, −24, ...

Comparing Sequences

13. Explain how arithmetic and geometric graphs differ.

14. Compare the sequences 2, 4, 6, 8, ... and 2, 4, 8, 16, ...

15. Which grows faster for large n: aₙ = 3n or bₙ = 3ⁿ?

16. Explain why a geometric sequence with r = 1 produces a horizontal graph.

Applications

17. A bacterial population begins with 50 cells and doubles every hour. Find the expected population after five hours.

18. An investment of $1,000 grows by 10% annually. Write its value after three years.

19. A vehicle worth $20,000 loses 20% of its value each year. Find its value after four years.

20. A radioactive sample begins with 640 mg and halves every five years. Find the expected remaining mass after 15 years.

Answer Key

Extended Challenge

Creating Tables and Graphs

1. Create a table and graph for the first six terms of aₙ = 4 × 2ⁿ⁻¹.

2. Create a table and graph for the first six terms of aₙ = 128(0.5)ⁿ⁻¹.

3. Graph the first six terms of 3, −6, 12, −24, ...

4. Graph the first six terms of 5, 5, 5, 5, ...

5. Compare the graphs of aₙ = 2 × 3ⁿ⁻¹ and bₙ = 2 × 2ⁿ⁻¹.

Finding Formulas from Graph Data

6. Find the nth term formula for points (1, 3), (2, 9), (3, 27).

7. Find the nth term formula for points (1, 100), (2, 50), (3, 25).

8. A geometric sequence graph contains (2, 12) and (5, 324). Find the common ratio and nth term formula.

9. A geometric sequence has a first term of 5 and a common ratio of −2. Write its formula and graph the first five terms.

10. A geometric sequence has a first term of 200 and a common ratio of 0.8. Graph the first six terms.

Comparing Linear and Exponential Growth

11. Compare the first eight terms of aₙ = 5n and bₙ = 5 × 2ⁿ⁻¹.

12. Determine the first term position where bₙ = 3ⁿ exceeds aₙ = 10n.

13. Explain why exponential growth can eventually exceed linear growth.

14. Explain why exponential decay differs from a constant decrease.

15. Describe how changing the common ratio affects the shape of a geometric sequence graph.

Real-World Applications

16. A bacterial population begins with 200 cells and triples every hour. Graph the expected population over the first five hours.

17. An investment begins at $5,000 and grows by 6% annually. Graph its value over the first six years.

18. A radioactive sample begins with 1,024 mg and halves every four years. Graph the expected remaining mass over 20 years at four-year intervals.

19. A vehicle worth $40,000 loses 15% of its value annually. Graph its value over the first five years.

20. A ball rebounds to 75% of its previous maximum height. If dropped from 8 metres, graph the maximum heights after the first five bounces, including the initial height.

Final Challenge: Comparing Growth and Decay

Two geometric sequences are defined by:

Sequence A: aₙ = 5 × 2ⁿ⁻¹

Sequence B: bₙ = 160(0.5)ⁿ⁻¹

Complete the investigation.

1. Identify the first term of Sequence A.

2. Identify the first term of Sequence B.

3. Determine the common ratio of each sequence.

4. Create a table for the first eight terms of both sequences.

5. Write the coordinates for the first five terms of each sequence.

6. Plot both sequences on the same coordinate plane.

7. Describe the graphical pattern of Sequence A.

8. Describe the graphical pattern of Sequence B.

9. Explain why Sequence A represents exponential growth.

10. Explain why Sequence B represents exponential decay.

11. Determine the third term of each sequence.

12. Determine the fifth term of each sequence.

13. Calculate the tenth term of each sequence.

14. Explain how the common ratios affect the graph shapes.

15. Compare the vertical changes between consecutive points.

16. Explain why the graphs are discrete.

17. Determine whether the sequences ever have equal values at the same positive integer term position.

18. Describe a real-world situation that could be modelled by Sequence A.

19. Describe a real-world situation that could be modelled by Sequence B.

20. Explain why understanding exponential growth and decay is important in mathematics and science.

Further Investigation: Design Your Own Geometric Graphs

Create three geometric sequences:

• Sequence A: Exponential growth with a positive common ratio greater than 1.

• Sequence B: Exponential decay with a positive common ratio between 0 and 1.

• Sequence C: Alternating signs with a negative common ratio.

For each sequence:

1. Identify the first term.

2. Identify the common ratio.

3. Write the nth term formula.

4. Generate the first eight terms.

5. Create a table of values.

6. Write the ordered pairs.

7. Plot the sequence on a coordinate plane.

8. Describe the graphical pattern.

9. Predict the tenth term.

10. Explain how the graph differs from an arithmetic sequence.

11. Identify a possible real-world application.

12. Explain any limitations of the model.

4. Geometric Series

Geometric Series

Introduction

A geometric series is the sum of the terms in a geometric sequence.

A geometric sequence is an ordered list of numbers in which each term is obtained by multiplying the previous term by a constant value called the common ratio.

For example, consider the geometric sequence:

2, 4, 8, 16, 32

Each term is twice the previous term.

When we add these terms together, we create a geometric series:

2 + 4 + 8 + 16 + 32 = 62

The distinction is important:

• A geometric sequence is a list of terms.

• A geometric series is the addition of those terms.

Geometric series appear in many mathematical and real-world situations, including compound interest, savings plans, population growth, repeated discounts, bouncing balls, and patterns involving repeated multiplication.

Understanding geometric series allows us to calculate the total accumulated value of a geometric pattern rather than just one individual term.

What Is a Geometric Sequence?

A geometric sequence is a list of numbers in which the ratio between consecutive terms is constant.

Consider:

3, 9, 27, 81, 243, ...

Each term is multiplied by 3.

3 × 3 = 9

9 × 3 = 27

27 × 3 = 81

81 × 3 = 243

The common ratio is:

r = 3

The nth term formula is:

aₙ = a₁rⁿ⁻¹

where:

• aₙ is the nth term.

• a₁ is the first term.

• r is the common ratio.

• n is the term number.

For this sequence:

a₁ = 3

r = 3

Therefore:

aₙ = 3 × 3ⁿ⁻¹

This formula calculates individual terms in the sequence.

What Is a Geometric Series?

A geometric series is formed by adding terms from a geometric sequence.

Consider the sequence:

3, 9, 27, 81, 243

The corresponding series is:

3 + 9 + 27 + 81 + 243

Calculate the total:

3 + 9 = 12

12 + 27 = 39

39 + 81 = 120

120 + 243 = 363

Therefore:

3 + 9 + 27 + 81 + 243 = 363

The sum of the first five terms is:

S₅ = 363

Important Notation

The letter S represents a sum.

The subscript tells us how many terms are included.

For example:

• S₁ represents the first term.

• S₂ represents the sum of the first two terms.

• S₃ represents the sum of the first three terms.

• S₅ represents the sum of the first five terms.

• Sₙ represents the sum of the first n terms.

This notation is useful when working with longer series.

Distinguishing Between Sequences and Series

Sequences and series are closely related, but they describe different mathematical ideas.

Sequence Example

5, 10, 20, 40, 80

This is a geometric sequence.

It lists five terms.

Series Example

5 + 10 + 20 + 40 + 80

This is a geometric series.

It adds the five terms together.

Calculate:

5 + 10 + 20 + 40 + 80 = 155

Therefore:

S₅ = 155

Comparison

Important Observation

A sequence and its corresponding series contain the same individual terms.

The difference is whether we are listing the terms or adding them.

Understanding the Common Ratio

The common ratio is the fixed multiplier between consecutive terms of a geometric sequence.

It is represented by:

r

For consecutive nonzero terms:

r = a₂/a₁

Worked Example

Consider:

4, 12, 36, 108, ...

Calculate:

12/4 = 3

36/12 = 3

108/36 = 3

Therefore:

r = 3

The corresponding series is:

4 + 12 + 36 + 108

Why Does the Common Ratio Matter?

The common ratio determines how the terms change.

If r > 1 and the first term is positive, the terms increase.

If 0 < r < 1 and the first term is positive, the terms decrease.

If r < 0 and the first term is nonzero, the terms alternate in sign.

The common ratio helps us identify geometric patterns and generate the terms needed for a series.

Calculating a Geometric Series by Direct Addition

The simplest way to calculate the sum of a finite geometric series is to add its terms directly.

Worked Example

Find the sum:

2 + 6 + 18 + 54

Step 1: Identify the Terms

The terms are:

2, 6, 18, 54

Step 2: Confirm the Pattern

6/2 = 3

18/6 = 3

54/18 = 3

The common ratio is 3.

Step 3: Add the Terms

2 + 6 = 8

8 + 18 = 26

26 + 54 = 80

Final Answer

S₄ = 80

This is a finite geometric series because it contains exactly four terms.

Another Example: A Five-Term Series

Find the sum:

3 + 6 + 12 + 24 + 48

Step 1: Identify the Common Ratio

6/3 = 2

12/6 = 2

24/12 = 2

Therefore:

r = 2

Step 2: Add the Terms

3 + 6 = 9

9 + 12 = 21

21 + 24 = 45

45 + 48 = 93

Final Answer

S₅ = 93

Checking the Result

We can check using a calculator:

3 + 6 + 12 + 24 + 48 = 93

The result is correct.

Finite Geometric Series

A finite geometric series contains a limited number of terms.

For example:

1 + 2 + 4 + 8 + 16

This series has five terms.

Its sum is:

1 + 2 + 4 + 8 + 16 = 31

Therefore:

S₅ = 31

Another Example

5 + 15 + 45 + 135

This series contains four terms.

Calculate:

5 + 15 + 45 + 135 = 200

Therefore:

S₄ = 200

Important Observation

A finite geometric series always has a calculable sum, provided its terms are finite real numbers.

The sum may be positive, negative, or zero.

Infinite Geometric Series

An infinite geometric series continues without a final term.

For example:

1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

The dots indicate that the pattern continues indefinitely.

Partial Sums

Although the series has infinitely many terms, we can calculate the sums of its first few terms.

S₁ = 1

S₂ = 1 + 1/2 = 1.5

S₃ = 1 + 1/2 + 1/4 = 1.75

S₄ = 1 + 1/2 + 1/4 + 1/8 = 1.875

S₅ = 1 + 1/2 + 1/4 + 1/8 + 1/16 = 1.9375

Table of Partial Sums

The sums approach 2.

This is an example of a convergent geometric series.

Why Is This Interesting?

An infinite series can sometimes approach a finite value even though it contains infinitely many terms.

However, not all infinite geometric series have finite sums.

For example:

1 + 2 + 4 + 8 + 16 + ...

continues increasing without bound.

Exploring Partial Sums

The following illustration shows how the partial sums of a geometric series change as more terms are added.

For a geometric series with first term a and common ratio r, the infinite sum is finite when |r| < 1.

In that case:

S∞ = a/(1 − r)

For the example above:

a = 1

r = 1/2

S∞ = 1/(1 − 1/2)

S∞ = 2

This formula is introduced here to show why some infinite series have finite totals. The main focus of these notes remains calculating finite geometric series by direct addition.

Partial Sums of a Geometric Series

A partial sum is the sum of a specified number of terms from a series.

Consider:

2 + 4 + 8 + 16 + 32 + ...

First Partial Sum

S₁ = 2

Second Partial Sum

S₂ = 2 + 4 = 6

Third Partial Sum

S₃ = 2 + 4 + 8 = 14

Fourth Partial Sum

S₄ = 2 + 4 + 8 + 16 = 30

Fifth Partial Sum

S₅ = 2 + 4 + 8 + 16 + 32 = 62

Table

Important Observation

Each new partial sum is found by adding the next term to the previous partial sum.

Therefore:

Sₙ = Sₙ₋₁ + aₙ, for n ≥ 2

This relationship is useful for calculating sums step by step.

Representing Geometric Series Using Mathematical Notation

Mathematicians use symbols to represent series efficiently.

Expanded Notation

A geometric series can be written as:

a₁ + a₂ + a₃ + ... + aₙ

For example:

2 + 6 + 18 + 54 + 162

Using the Common Ratio

Because each term is multiplied by r, we can write:

a₁ + a₁r + a₁r² + a₁r³ + ... + a₁rⁿ⁻¹

For example, if a₁ = 2 and r = 3:

2 + 2(3) + 2(3²) + 2(3³) + 2(3⁴)

This becomes:

2 + 6 + 18 + 54 + 162

Using Sₙ Notation

The sum of the first n terms is represented by:

Sₙ

For example:

S₅ = 2 + 6 + 18 + 54 + 162

S₅ = 242

Why Is Mathematical Notation Useful?

Mathematical notation allows us to:

• Describe patterns clearly.

• Represent many terms efficiently.

• Avoid writing extremely long expressions.

• Communicate mathematical ideas precisely.

• Develop formulas for calculating sums.

Introduction to Sigma Notation

The Greek capital letter sigma, Σ, is used to represent summation.

It means "add together."

For example:

Σ from k = 1 to 4 of 2ᵏ

means:

2¹ + 2² + 2³ + 2⁴

Calculate:

2 + 4 + 8 + 16 = 30

Therefore, the sum is 30.

Understanding the Parts

In sigma notation:

• Σ means summation.

• k is the index.

• The lower value tells us where to start.

• The upper value tells us where to stop.

• The expression tells us what to calculate for each index.

Another Example

Consider:

Σ from k = 0 to 3 of 3 × 2ᵏ

Expand:

3 × 2⁰ + 3 × 2¹ + 3 × 2² + 3 × 2³

Calculate:

3 + 6 + 12 + 24

Sum = 45

Important Observation

Sigma notation is a compact way to write a series.

It does not change the values being added.

Comparing Arithmetic and Geometric Series

Arithmetic and geometric series are both formed by adding terms from sequences.

However, their underlying patterns differ.

Arithmetic Series

Consider:

2 + 5 + 8 + 11 + 14

The corresponding arithmetic sequence has a common difference of 3.

Calculate:

2 + 5 + 8 + 11 + 14 = 40

Geometric Series

Consider:

2 + 6 + 18 + 54 + 162

The corresponding geometric sequence has a common ratio of 3.

Calculate:

2 + 6 + 18 + 54 + 162 = 242

Comparison

Important Conclusion

The type of series depends on the pattern of the terms being added.

Geometric Series with Fractions

Geometric series may contain fractions.

Worked Example

Find the sum:

1/2 + 1/4 + 1/8 + 1/16

Step 1: Identify the Common Ratio

(1/4) ÷ (1/2) = 1/2

Therefore:

r = 1/2

Step 2: Use a Common Denominator

1/2 = 8/16

1/4 = 4/16

1/8 = 2/16

1/16 = 1/16

Step 3: Add

8/16 + 4/16 + 2/16 + 1/16

= 15/16

Final Answer

S₄ = 15/16

Decimal Form

15/16 = 0.9375

Therefore:

S₄ = 0.9375

Geometric Series with Decimal Values

Worked Example

Find the sum:

10 + 5 + 2.5 + 1.25

Step 1: Identify the Common Ratio

5/10 = 0.5

2.5/5 = 0.5

1.25/2.5 = 0.5

Therefore:

r = 0.5

Step 2: Add the Terms

10 + 5 = 15

15 + 2.5 = 17.5

17.5 + 1.25 = 18.75

Final Answer

S₄ = 18.75

Interpretation

The terms decrease, but the partial sums increase because every term is positive.

This is an important distinction between the behaviour of a sequence and the behaviour of its partial sums.

Geometric Series with Negative Terms

Geometric series can contain negative terms.

Worked Example

Find the sum:

3 − 6 + 12 − 24 + 48

The corresponding sequence is:

3, −6, 12, −24, 48

The common ratio is:

r = −2

Step 1: Add the Terms Carefully

3 − 6 = −3

−3 + 12 = 9

9 − 24 = −15

−15 + 48 = 33

Final Answer

S₅ = 33

Important Observation

A negative common ratio produces alternating positive and negative terms.

The partial sums may increase or decrease depending on the next term.

Another Example

Find:

8 − 4 + 2 − 1

Calculate:

8 − 4 = 4

4 + 2 = 6

6 − 1 = 5

Therefore:

S₄ = 5

Real-World Application: Saving Money

Suppose a student saves $5 during the first week.

Each following week, the student saves twice as much as the previous week.

Weekly Savings

The amounts saved each week form a geometric sequence.

Total Savings

The total saved after five weeks is a geometric series:

5 + 10 + 20 + 40 + 80

Calculate:

5 + 10 = 15

15 + 20 = 35

35 + 40 = 75

75 + 80 = 155

Final Answer

Total savings = $155

Interpretation

The sequence represents the amount saved each week.

The series represents the total accumulated savings.

This distinction is useful in financial planning.

Real-World Application: Bacterial Growth

Suppose a bacterial culture begins with 100 cells.

The population doubles every hour.

Population Values

The population values form a geometric sequence.

A Series Interpretation

Suppose four separate samples contain 100, 200, 400, and 800 cells.

The total number of cells across the four samples is:

100 + 200 + 400 + 800

= 1,500 cells

Important Distinction

Adding population measurements taken at different times does not tell us the population at the final time.

The population after three hours is 800 cells, not 1,500.

The series is meaningful only when the quantities being added represent separate contributions or another clearly defined total.

Real-World Application: A Bouncing Ball

A ball is dropped from a height of 4 metres.

After each bounce, it rises to half of its previous maximum height.

Maximum Heights

Initial height: 4 m

First rebound: 2 m

Second rebound: 1 m

Third rebound: 0.5 m

Fourth rebound: 0.25 m

The rebound heights form a geometric sequence.

Sum of Rebound Heights

Suppose we want the sum of the maximum heights reached during the first four rebounds.

2 + 1 + 0.5 + 0.25

= 3.75 m

Important Distinction

This sum represents the total of the four rebound heights.

It is not the total distance travelled by the ball.

To calculate the distance travelled, we must account for both upward and downward motion.

Total Distance Through Four Complete Rebounds

Initial drop:

4 m

First rebound up and down:

2 + 2 = 4 m

Second rebound up and down:

1 + 1 = 2 m

Third rebound up and down:

0.5 + 0.5 = 1 m

Fourth rebound up and down:

0.25 + 0.25 = 0.5 m

Total distance:

4 + 4 + 2 + 1 + 0.5

Total = 11.5 m

This assumes the ball completes its fourth rebound and returns to the ground.

Real-World Application: Repeated Discounts

Suppose a product initially costs $200.

Its price is reduced by 10% each week.

The price retains 90% of its previous value.

Therefore:

r = 0.9

Price Values

The prices form a geometric sequence.

A Series Interpretation

Suppose four identical products are purchased separately at these four prices.

The total cost is:

200 + 180 + 162 + 145.80

Total = $687.80

Interpretation

The sequence represents the changing price.

The series represents the total amount paid for four purchases.

This is another example of how a sequence and a series describe different quantities.

Real-World Application: A Geometric Pattern of Objects

Suppose a display contains:

• 2 objects in the first row.

• 4 objects in the second row.

• 8 objects in the third row.

• 16 objects in the fourth row.

• 32 objects in the fifth row.

Sequence

2, 4, 8, 16, 32

Series

2 + 4 + 8 + 16 + 32

Calculate the Total

2 + 4 = 6

6 + 8 = 14

14 + 16 = 30

30 + 32 = 62

Final Answer

62 objects

Interpretation

The sequence describes the number of objects in each row.

The series describes the total number of objects in the display.

Organising Geometric Series Problems

When solving a geometric series problem, follow a consistent process.

Step 1: Identify the Sequence

Determine the individual terms.

Step 2: Check the Common Ratio

Divide consecutive nonzero terms.

Step 3: Determine the Number of Terms

Identify how many terms must be added.

Step 4: Write the Series

Use addition signs between the terms.

Step 5: Add Carefully

Calculate the total.

Step 6: Check the Result

Verify the calculation using mental arithmetic, written methods, or a calculator.

Step 7: Interpret the Answer

Include units when solving a real-world problem.

For example:

• Dollars for savings.

• Metres for distances.

• Objects for counting problems.

• Cells for biological samples.

Common Mistakes

Confusing a sequence with a series

A sequence lists terms.

A series adds terms.

Finding the last term instead of the sum

The last term and the total sum are different quantities.

For example:

2 + 4 + 8 + 16

Last term = 16

Sum = 30

Using the common difference instead of the common ratio

Geometric sequences involve repeated multiplication.

Forgetting a term

When adding a series, include every required term exactly once.

Making arithmetic errors

Large terms can make direct addition difficult.

Check calculations carefully.

Ignoring negative signs

Alternating geometric series may contain positive and negative terms.

Confusing partial sums with individual terms

aₙ represents a term.

Sₙ represents a sum.

Assuming an infinite series always has a finite sum

Some infinite geometric series converge, while others diverge.

Adding measurements that should not be added

For example, adding a population measured at different times does not give the final population.

Forgetting units

Real-world answers should include appropriate units.

Key Terms

• Sequence: Ordered list of numbers following a pattern.

• Series: Sum of the terms of a sequence.

• Geometric Sequence: Sequence with a constant ratio between consecutive terms.

• Geometric Series: Sum of terms from a geometric sequence.

• Term: Individual number in a sequence.

• First Term: Value at the beginning of a sequence.

• Common Ratio: Fixed multiplier between consecutive terms.

• Finite Series: Series containing a limited number of terms.

• Infinite Series: Series continuing indefinitely.

• Sum: Result of adding numbers together.

• Partial Sum: Sum of a specified number of terms.

• Sₙ: Sum of the first n terms.

• aₙ: nth term of a sequence.

• Sigma Notation: Compact mathematical notation used to represent summation.

• Summation: Process of adding terms together.

• Index: Variable used to identify terms in summation notation.

• Convergent Series: Infinite series whose partial sums approach a finite limit.

• Divergent Series: Infinite series whose partial sums do not approach a finite limit.

• Arithmetic Series: Sum of terms from an arithmetic sequence.

• Alternating Series: Series containing terms that alternate in sign.

• Accumulation: Process of building a total through repeated additions.

• Mathematical Model: Mathematical representation of a real-world situation.

Key Takeaways

• A geometric sequence is a list of terms with a constant ratio.

• A geometric series is the sum of terms from a geometric sequence.

• A sequence lists values, while a series adds them.

• The common ratio is represented by r.

• The nth term of a geometric sequence is aₙ = a₁rⁿ⁻¹.

• The sum of the first n terms is represented by Sₙ.

• A finite geometric series contains a fixed number of terms.

• Finite geometric series can be calculated by direct addition.

• Partial sums show how a total changes as more terms are included.

• Geometric series may contain positive terms, negative terms, fractions, or decimals.

• Sigma notation provides a compact way to represent sums.

• Some infinite geometric series approach finite values.

• Geometric series can model accumulated savings, repeated contributions, and totals in geometric patterns.

• Real-world problems require careful interpretation of what the terms represent.

Check Your Understanding

Sequences and Series

1. Define a geometric sequence.

2. Define a geometric series.

3. Explain the difference between a sequence and a series.

4. Convert the sequence 3, 6, 12, 24 into a series.

Identifying Common Ratios

5. Find the common ratio of 2, 8, 32, 128.

6. Find the common ratio of 100, 50, 25, 12.5.

7. Find the common ratio of 3, −9, 27, −81.

8. Determine whether 2, 5, 8, 11 is geometric.

Calculating Finite Sums

9. Calculate 2 + 4 + 8 + 16.

10. Calculate 3 + 9 + 27 + 81.

11. Calculate 5 + 10 + 20 + 40 + 80.

12. Calculate 100 + 50 + 25 + 12.5.

13. Calculate 4 − 8 + 16 − 32.

14. Calculate 1/2 + 1/4 + 1/8.

Mathematical Notation

15. What does S₅ represent?

16. What does the sigma symbol Σ mean?

17. Write 2 + 6 + 18 + 54 using Sₙ notation.

Applications

18. A student saves $10, $20, $40, and $80 over four weeks. Find the total savings.

19. A display contains 3, 9, 27, and 81 objects in successive rows. Find the total number of objects.

20. Explain why adding population measurements taken at different times does not necessarily give a meaningful total population.

Answer Key

Extended Challenge

Identifying Series

1. Convert 4, 12, 36, 108 into a geometric series.

2. Convert 100, 50, 25, 12.5 into a geometric series.

3. Identify the first term and common ratio of 5 + 15 + 45 + 135.

4. Explain why 2 + 5 + 8 + 11 is not a geometric series.

5. Write the first five terms of a geometric series with a₁ = 3 and r = 4.

Calculating Sums

6. Calculate 2 + 6 + 18 + 54 + 162.

7. Calculate 4 + 8 + 16 + 32 + 64.

8. Calculate 81 + 27 + 9 + 3 + 1.

9. Calculate 5 − 10 + 20 − 40 + 80.

10. Calculate 1 + 1/2 + 1/4 + 1/8 + 1/16.

Partial Sums

11. Find S₃ for 3 + 6 + 12 + 24 + ...

12. Find S₄ for 5 + 15 + 45 + 135 + ...

13. Find S₅ for 2 + 4 + 8 + 16 + 32 + ...

14. Find S₆ for 1 + 3 + 9 + 27 + 81 + 243 + ...

15. Explain how Sₙ differs from aₙ.

Real-World Applications

16. A student saves $5 in the first week and doubles the amount saved each following week. Find the total saved over six weeks.

17. A display has 2 objects in its first row and triples the number in each successive row. Find the total number of objects in the first five rows.

18. A ball rebounds to half of its previous height. If the first rebound reaches 4 metres, find the sum of the first five rebound heights.

19. Four identical products are purchased at prices of $100, $80, $64, and $51.20. Find the total cost.

20. A company awards bonuses of $50, $100, $200, and $400 to four different employees. Find the total amount awarded.

Final Challenge: Investigating a Geometric Series

Consider the geometric sequence:

4, 12, 36, 108, 324, ...

Complete the investigation.

1. Identify the first term.

2. Determine the common ratio.

3. Explain why the sequence is geometric.

4. Write the corresponding geometric series.

5. Calculate S₁.

6. Calculate S₂.

7. Calculate S₃.

8. Calculate S₄.

9. Calculate S₅.

10. Create a table showing the first five partial sums.

11. Explain how the partial sums change as more terms are added.

12. Write the nth term formula.

13. Calculate the sixth term.

14. Calculate the sum of the first six terms.

15. Explain the difference between a₆ and S₆.

16. Write the first five terms using sigma notation.

17. Compare this series with the arithmetic series 4 + 8 + 12 + 16 + 20.

18. Calculate the sum of the arithmetic series.

19. Explain why the geometric series grows more rapidly.

20. Describe a real-world situation that could be represented by this geometric series.

Further Investigation: Create Your Own Geometric Series

Create three geometric series:

• Series A: A geometric series with increasing positive terms.

• Series B: A geometric series with decreasing positive terms.

• Series C: A geometric series with alternating positive and negative terms.

For each series:

1. Identify the first term.

2. Identify the common ratio.

3. Generate the first six terms.

4. Write the terms as a series.

5. Calculate S₁.

6. Calculate S₂.

7. Calculate S₃.

8. Calculate S₄.

9. Calculate S₅.

10. Calculate S₆.

11. Create a table of partial sums.

12. Describe how the partial sums behave.

13. Write the nth term formula.

14. Explain how the series differs from its corresponding sequence.

15. Identify a possible real-world interpretation.

5. Applications of Geometric Growth

Applications of Geometric Growth

Introduction

Geometric growth and decay occur when a quantity changes by the same multiplicative factor over equal intervals.

Unlike arithmetic growth, which involves adding a fixed amount, geometric growth involves multiplying by a fixed number.

For example, consider two populations that both begin with 100 organisms.

• Population A increases by 50 organisms each year.

• Population B increases by 50% each year.

After five years, the populations will be very different because one follows an arithmetic pattern while the other follows a geometric pattern.

Geometric relationships appear in many real-world situations, including population growth, compound interest, inflation, depreciation, radioactive decay, repeated discounts, and biological reproduction.

Understanding these patterns allows us to predict future values, compare alternatives, and calculate accumulated totals.

Recognising Geometric Growth and Decay

A geometric sequence has a constant ratio between consecutive terms.

Consider:

4, 8, 16, 32, 64, ...

Each term is multiplied by 2.

The common ratio is:

r = 2

This is geometric growth.

Now consider:

160, 80, 40, 20, 10, ...

Each term is multiplied by 0.5.

The common ratio is:

r = 0.5

This is geometric decay.

General Formula

For a sequence whose first term is at position n = 1:

aₙ = a₁rⁿ⁻¹

where:

• aₙ = value of the nth term

• a₁ = first term

• r = common ratio

• n = term number

For situations measured from an initial time of zero, a more convenient formula is:

A(t) = A₀rᵗ

where:

• A(t) = quantity after t time intervals

• A₀ = initial quantity

• r = multiplier per interval

• t = number of intervals elapsed

The formulas describe the same repeated-multiplication pattern, but use different starting positions.

Growth, Decay, and Constant Patterns

For populations, money balances, and many physical quantities, negative multipliers are usually not meaningful.

Converting Percentage Changes into Multipliers

Many geometric problems describe changes using percentages rather than common ratios.

The first step is to convert the percentage into a decimal multiplier.

Percentage Increase

For an increase of p%:

Growth multiplier = 1 + p/100

Examples:

A 25% increase means the new value is 125% of the previous value.

Therefore:

New value = Previous value × 1.25

Percentage Decrease

For a decrease of p%:

Decay multiplier = 1 − p/100

Examples:

A 20% decrease means the new value is 80% of the previous value.

Therefore:

New value = Previous value × 0.80

Worked Example

A quantity increases by 12% each year.

Find the multiplier.

r = 1 + 12/100

r = 1 + 0.12

r = 1.12

If the quantity starts at 500:

A(t) = 500(1.12)ᵗ

This formula models a 12% increase every year.

Modelling Population Growth

Population growth is one of the most common applications of geometric sequences.

Under ideal conditions, some populations increase by approximately the same percentage during each time interval.

Examples include:

• Bacterial reproduction.

• Insect populations.

• Plant populations during favourable growing conditions.

• Introduced species during early population expansion.

Worked Example: Bacterial Growth

A bacterial culture begins with 200 cells.

The population doubles every hour.

Step 1: Identify the Initial Population

P₀ = 200 cells

Step 2: Identify the Multiplier

The population doubles.

r = 2

Step 3: Write the Formula

P(t) = 200 × 2ᵗ

where t is the number of hours elapsed.

Step 4: Create a Table

Step 5: Predict the Population After Eight Hours

P(8) = 200 × 2⁸

P(8) = 200 × 256

P(8) = 51,200 cells

Interpretation

The population doubles every hour, so the number of new cells produced during each hour increases as the population grows.

The increase from 200 to 400 is only 200 cells.

The increase from 6,400 to 12,800 is 6,400 cells.

The percentage growth remains constant even though the numerical increase becomes larger.

Population Growth Using Percentages

Not all populations double or triple.

Some increase by smaller percentages.

Worked Example

A town has an initial population of 12,000 people.

Its population grows by 3% each year.

Estimate the population after five years.

Step 1: Identify the Initial Population

P₀ = 12,000

Step 2: Convert the Percentage

r = 1 + 0.03

r = 1.03

Step 3: Write the Model

P(t) = 12,000(1.03)ᵗ

Step 4: Substitute t = 5

P(5) = 12,000(1.03)⁵

P(5) ≈ 12,000 × 1.159274

P(5) ≈ 13,911 people

Interpretation

The model predicts approximately 13,911 people after five years.

Because population is counted in whole people, we round the result to the nearest whole number.

Limitations of Population Models

Real populations do not necessarily grow geometrically forever.

Growth can be affected by:

• Food and water availability.

• Disease.

• Competition.

• Migration.

• Predation.

• Birth and death rates.

• Environmental conditions.

Geometric models are often most useful over periods when the percentage growth rate is reasonably stable.

When resources become limited, population growth may slow and approach a carrying capacity rather than continuing exponentially.

Understanding Compound Interest

Compound interest is another important application of geometric growth.

When money earns compound interest, interest is calculated on the current balance, including interest earned during previous periods.

This creates repeated multiplication.

Simple Interest Versus Compound Interest

Suppose $1,000 earns 10% interest annually.

With simple interest, the same $100 is added each year.

With compound interest, each year earns 10% of the updated balance.

Simple interest follows an arithmetic pattern.

Compound interest follows a geometric pattern.

Why Does Compound Interest Grow Faster?

With compound interest, the interest earned becomes part of the balance used to calculate future interest.

This is sometimes described as earning interest on interest.

Compound Interest Formula

For an initial investment with a fixed interest rate compounded once per period:

FV = PV(1 + i)ⁿ

where:

• FV = future value

• PV = present value or initial investment

• i = interest rate per compounding period, expressed as a decimal

• n = number of compounding periods

This is a geometric model with multiplier:

r = 1 + i

Worked Example: Investment Growth

An investor deposits $2,500 into an account earning 6% annual interest, compounded annually.

Calculate the balance after eight years.

Step 1: Identify the Values

PV = $2,500

i = 0.06

n = 8

Step 2: Substitute

FV = 2,500(1.06)⁸

Step 3: Calculate

(1.06)⁸ ≈ 1.593848

FV ≈ 2,500 × 1.593848

FV ≈ $3,984.62

Step 4: Calculate the Interest Earned

Interest = Final balance − Initial investment

Interest = 3,984.62 − 2,500

Interest earned ≈ $1,484.62

Interpretation

The account grows because each year's interest is calculated using the balance accumulated during the previous years.

This model assumes a fixed interest rate, no additional deposits or withdrawals, and no fees or taxes.

How Compounding Frequency Changes Growth

Interest may be compounded annually, semiannually, quarterly, monthly, or daily.

When a nominal annual interest rate is divided into multiple compounding periods, the formula becomes:

A = P(1 + i/m)ᵐᵗ

where:

• P = initial principal

• i = nominal annual interest rate

• m = number of compounding periods per year

• t = time in years

Worked Example

An investment of $1,000 earns 12% nominal annual interest, compounded monthly.

Find its value after one year.

P = 1,000

i = 0.12

m = 12

t = 1

Substitute:

A = 1,000(1 + 0.12/12)¹²

A = 1,000(1.01)¹²

A ≈ $1,126.83

The investment earns approximately $126.83 in interest.

This is slightly more than the $120 earned if 12% interest were compounded only once at the end of the year.

Exploring Compound Interest

The following model allows you to explore how the initial investment, interest rate, and number of periods affect compound growth.

What Can We Learn?

Changing the initial investment changes the starting value.

Changing the interest rate changes the common ratio.

Changing the number of years changes how many times the balance is multiplied.

Even a small change in the interest rate can produce a significant difference over long periods.

Geometric Decay and Depreciation

Geometric relationships also describe situations in which a quantity decreases by a fixed percentage during each interval.

Depreciation is a decrease in an asset's value over time.

For example, a vehicle may lose a percentage of its remaining value each year.

Worked Example: Vehicle Depreciation

A vehicle is purchased for $30,000.

Its value decreases by 15% each year.

Find its estimated value after four years.

Step 1: Identify the Initial Value

V₀ = $30,000

Step 2: Calculate the Multiplier

r = 1 − 0.15

r = 0.85

Step 3: Write the Formula

V(t) = 30,000(0.85)ᵗ

Step 4: Calculate

V(4) = 30,000(0.85)⁴

V(4) = 30,000 × 0.52200625

V(4) ≈ $15,660.19

Step 5: Calculate the Total Depreciation

30,000 − 15,660.19 = 14,339.81

Total depreciation ≈ $14,339.81

Interpretation

The vehicle loses 15% of its remaining value each year, not 15% of its original purchase price.

Therefore, the amount lost in dollars becomes smaller over time.

Actual resale values may differ from this simplified model.

Repeated Percentage Changes

Geometric models are especially useful when percentages are applied repeatedly.

Worked Example: Repeated Discounts

A jacket originally costs $200.

The store reduces its price by 20%, then applies an additional 10% discount.

First Discount

200 × 0.80 = 160

New price = $160

Second Discount

160 × 0.90 = 144

Final price = $144

Overall Discount

200 − 144 = 56

56/200 × 100% = 28%

The total discount is 28%, not 30%.

Why?

The second discount is applied to the already reduced price.

Repeated percentage changes must be multiplied, not simply added.

The combined multiplier is:

0.80 × 0.90 = 0.72

Therefore, the jacket retains 72% of its original price.

Another Example: Increase Followed by Decrease

A product increases in price by 20%, then decreases by 20%.

The combined multiplier is:

1.20 × 0.80 = 0.96

The final price is 96% of the original price.

The overall change is a 4% decrease.

Equal percentage increases and decreases do not cancel each other because they are applied to different starting values.

Geometric Series in Real-World Problems

So far, we have used geometric sequences to find individual values.

However, many practical problems require the sum of several values.

A geometric series adds the terms of a geometric sequence.

For example:

5 + 10 + 20 + 40 + 80

The corresponding sequence has:

a₁ = 5

r = 2

The sum is:

5 + 10 + 20 + 40 + 80 = 155

Therefore:

S₅ = 155

Finite Geometric Series Formula

For r ≠ 1:

Sₙ = a₁(1 − rⁿ)/(1 − r)

An equivalent form is:

Sₙ = a₁(rⁿ − 1)/(r − 1)

where:

• Sₙ = sum of the first n terms

• a₁ = first term

• r = common ratio

• n = number of terms

If r = 1, every term is equal and:

Sₙ = na₁

Why Does the Formula Work?

Consider:

Sₙ = a₁ + a₁r + a₁r² + ... + a₁rⁿ⁻¹

Multiply by r:

rSₙ = a₁r + a₁r² + ... + a₁rⁿ

Subtract the second expression from the first:

Sₙ − rSₙ = a₁ − a₁rⁿ

Factor:

Sₙ(1 − r) = a₁(1 − rⁿ)

Divide by 1 − r:

Sₙ = a₁(1 − rⁿ)/(1 − r)

The formula works because the middle terms cancel during subtraction.

Application: Total Savings

A student saves $10 during the first week and doubles the amount saved each following week.

How much will the student save over eight weeks?

Step 1: Identify the Values

a₁ = 10

r = 2

n = 8

Step 2: Write the Formula

S₈ = 10(2⁸ − 1)/(2 − 1)

Step 3: Calculate

S₈ = 10(256 − 1)

S₈ = 10 × 255

S₈ = $2,550

Verification

The weekly savings are:

10, 20, 40, 80, 160, 320, 640, 1,280

Adding them gives:

$2,550

Interpretation

The eighth week's savings are $1,280.

The total savings over eight weeks are $2,550.

This illustrates the difference between a geometric sequence term and a geometric series sum.

Application: Regular Contributions to an Investment

Geometric series can also model a series of deposits that earn compound interest.

Suppose a person deposits $100 at the end of each year into an account earning 5% annually.

At the end of the fourth year, the deposits have the following values:

The total value is:

FV = 100(1.05)³ + 100(1.05)² + 100(1.05) + 100

FV = 115.7625 + 110.25 + 105 + 100

FV ≈ $431.01

The deposits form a geometric series when their values are measured at the same future date.

Important Distinction

A single initial deposit follows a geometric growth model.

A series of equal deposits can produce a geometric series when we calculate their accumulated future values.

Application: Bouncing Balls and Total Distance

A ball is dropped from a height of 8 metres.

After each bounce, it rises to 60% of its previous maximum height.

Step 1: Calculate the Rebound Heights

First rebound:

8 × 0.6 = 4.8 m

Second rebound:

4.8 × 0.6 = 2.88 m

Third rebound:

2.88 × 0.6 = 1.728 m

Fourth rebound:

1.728 × 0.6 = 1.0368 m

Step 2: Find the Total of the Rebound Heights

4.8 + 2.88 + 1.728 + 1.0368

= 10.4448 m

Step 3: Calculate the Total Distance

The initial drop is 8 metres.

Each complete rebound includes upward and downward motion.

Total distance through four complete rebounds:

D = 8 + 2(10.4448)

D = 28.8896 m

Interpretation

The ball travels approximately 28.89 metres by the time it returns to the ground after the fourth rebound.

The total distance is greater than the initial drop height because the ball repeatedly travels upward and downward.

Application: Radioactive Decay

Radioactive decay can be modelled using a geometric relationship when we measure the expected remaining quantity at equal intervals.

The half-life is the time required for half of a large sample's undecayed radioactive nuclei to decay, on average.

Worked Example

A radioactive sample initially contains 640 mg of an isotope.

Its half-life is 5 years.

Find the expected remaining mass after 20 years.

Step 1: Identify the Number of Half-Lives

20/5 = 4

Step 2: Identify the Multiplier

r = 0.5

Step 3: Write the Model

M = 640(0.5)ⁿ

where n is the number of completed half-life intervals.

Step 4: Calculate

M(4) = 640(0.5)⁴

M(4) = 640/16

M(4) = 40 mg

Interpretation

The model predicts that approximately 40 mg remains after 20 years.

The mass decreases geometrically because the expected remaining quantity is multiplied by 0.5 after every half-life.

Comparing Arithmetic and Geometric Growth

Suppose two investments begin at $1,000.

Investment A increases by $100 each year.

Investment B increases by 10% each year.

Arithmetic Model

A(t) = 1,000 + 100t

Geometric Model

B(t) = 1,000(1.10)ᵗ

Interpretation

The arithmetic model increases by a constant dollar amount.

The geometric model increases by a constant percentage.

Because the geometric model calculates growth from an increasing balance, its dollar increases become larger each year.

Choosing the Correct Mathematical Model

Not every situation involving change is geometric.

The key is identifying how the quantity changes.

Questions to Ask

When deciding which model to use, ask:

1. Does the quantity change by a constant amount or a constant percentage?

2. Is the change additive or multiplicative?

3. What is the starting value?

4. What is the common difference or common ratio?

5. What time interval does each step represent?

6. Are we finding an individual value or an accumulated total?

7. Is the model realistic over the required time period?

Common Mistakes

Adding a percentage instead of multiplying

A 10% increase means multiplying by 1.10, not adding 10 to the quantity.

Using the percentage as the multiplier

A 15% increase uses r = 1.15, not r = 0.15.

Using the wrong decay factor

A 20% decrease uses r = 0.80, not r = 0.20.

Confusing a sequence term with a series sum

A term represents one value. A series sum represents the total of several values.

Confusing the starting time

For models beginning at time zero, use an exponent of t. For sequences indexed from term one, use n − 1.

Adding repeated percentage changes

Successive percentage changes should be represented by multiplying their factors.

Rounding too early

Keep several decimal places during calculations and round the final answer appropriately.

Assuming geometric growth continues indefinitely

Real populations, investments, and physical systems may experience changing conditions.

Confusing simple and compound interest

Simple interest uses a fixed amount of interest each period. Compound interest uses the updated balance.

Using inconsistent time units

The multiplier and exponent must refer to the same interval. A monthly multiplier must be used with a number of months.

Key Terms

• Geometric Growth: Increase caused by repeated multiplication by a constant factor greater than 1.

• Geometric Decay: Decrease caused by repeated multiplication by a factor between 0 and 1.

• Geometric Sequence: Sequence with a constant ratio between consecutive terms.

• Geometric Series: Sum of terms from a geometric sequence.

• Common Ratio: Fixed multiplier between consecutive terms.

• Growth Factor: Multiplier used to represent a percentage increase.

• Decay Factor: Multiplier used to represent a percentage decrease.

• Initial Value: Quantity at the beginning of a model.

• Exponential Growth: Growth described by repeated multiplication.

• Exponential Decay: Decay described by repeated multiplication.

• Compound Interest: Interest calculated using the current balance, including previously earned interest.

• Simple Interest: Interest calculated only on the original principal.

• Principal: Initial amount invested or borrowed.

• Future Value: Value of an investment at a later time.

• Depreciation: Decrease in an asset's value over time.

• Inflation: General increase in prices over time.

• Half-Life: Time required for half of a radioactive sample's undecayed nuclei to decay on average.

• Partial Sum: Sum of a specified number of terms.

• Finite Geometric Series: Sum of a fixed number of geometric terms.

• Mathematical Model: Mathematical representation of a real-world process.

• Carrying Capacity: Maximum population an environment can sustainably support under given conditions.

• Accumulation: Building a total through repeated contributions.

• Percentage Change: Change expressed relative to the previous value.

Key Takeaways

• Geometric growth and decay involve repeated multiplication.

• A constant percentage change produces a geometric pattern.

• The multiplier for a p% increase is 1 + p/100.

• The multiplier for a p% decrease is 1 − p/100.

• Geometric sequences can model population growth, compound interest, and depreciation.

• The formula aₙ = a₁rⁿ⁻¹ calculates individual terms of a geometric sequence.

• The formula A(t) = A₀rᵗ models changes from an initial time of zero.

• Compound interest produces geometric growth because interest is calculated on the updated balance.

• Geometric series calculate accumulated totals.

• The finite geometric series formula is Sₙ = a₁(1 − rⁿ)/(1 − r), when r ≠ 1.

• Repeated percentage changes must be multiplied, not added.

• Arithmetic models involve constant amounts; geometric models involve constant ratios.

• Real-world models depend on assumptions and may not remain accurate indefinitely.

Check Your Understanding

Identifying Geometric Patterns

1. Explain the difference between arithmetic and geometric growth.

2. Determine the multiplier for a 12% increase.

3. Determine the multiplier for a 25% decrease.

4. Is a salary increase of $200 per year arithmetic or geometric?

5. Is a population increase of 8% per year arithmetic or geometric?

Population Growth

6. A population begins with 500 organisms and doubles each year. Write a formula for its population after t years.

7. Find the population after four years.

8. A town has 10,000 residents and grows by 2% annually. Write a geometric model.

9. Estimate its population after three years.

Compound Interest

10. An investment of $1,000 earns 5% compound interest annually. Write the growth formula.

11. Find its balance after three years.

12. Explain why compound interest produces geometric growth.

13. Explain the difference between simple and compound interest.

Repeated Percentage Changes

14. A product costs $100 and decreases in price by 20%. Find its new price.

15. The reduced price decreases by another 20%. Find the final price.

16. Explain why two successive 20% discounts do not produce a 40% discount.

Geometric Series

17. Calculate 5 + 10 + 20 + 40 + 80.

18. A student saves $10, $20, $40, and $80 over four weeks. Find the total.

19. Find the sum of the first five terms of 3, 6, 12, 24, ...

20. Explain the difference between a geometric sequence and a geometric series in a savings problem.

Answer Key

Extended Challenge

Growth and Decay Models

1. A population of 800 organisms grows by 15% annually. Write a formula and calculate the population after six years.

2. A bacterial culture begins with 150 cells and triples every two hours. Estimate the population after eight hours.

3. A city begins with 50,000 residents and grows by 1.8% annually. Estimate its population after ten years.

4. A vehicle worth $45,000 loses 12% of its value annually. Calculate its value after six years.

5. A radioactive sample initially contains 1,280 mg and has a half-life of four years. Calculate its expected remaining mass after 16 years.

Financial Applications

6. An investment of $5,000 earns 7% annual compound interest. Find its value after ten years.

7. Compare $2,000 invested for five years at 5% simple interest and 5% compound interest.

8. A product increases in price by 15%, then decreases by 10%. Determine the overall percentage change.

9. A computer costs $1,500 and depreciates by 20% annually. Find its estimated value after four years.

10. An investment earns 4% annually. Determine how many complete years are required for its balance to first exceed 150% of its initial value.

Geometric Series

11. A student saves $5 in the first week and doubles the amount saved each week. Find the total saved over ten weeks.

12. A theatre display contains 3 lights in its first row and twice as many in each successive row. Find the total lights in the first eight rows.

13. A ball rebounds to 80% of its previous height. If dropped from 10 metres, find the sum of its first five rebound heights.

14. Find the sum of the first six terms of 4, 12, 36, 108, ...

15. A company awards bonuses of $100, $150, $225, and $337.50. Find the total amount awarded.

Analysis and Evaluation

16. Explain why population growth cannot continue exponentially forever.

17. Compare the graphs of arithmetic and geometric growth.

18. Explain why a 50% increase followed by a 50% decrease does not return a quantity to its original value.

19. Describe a real-world situation in which a geometric series is more useful than a geometric sequence.

20. Explain why identifying the correct time interval is essential when applying geometric models.

Final Investigation: Planning a Savings Strategy

Two students begin saving money.

Student A saves $20 during the first week and increases the amount saved by $10 each following week.

Student B saves $20 during the first week and increases the amount saved by 25% each following week.

Complete the investigation.

1. Identify which savings plan follows an arithmetic sequence.

2. Identify which savings plan follows a geometric sequence.

3. Determine the common difference for Student A.

4. Determine the common ratio for Student B.

5. Write the nth term formula for each savings plan.

6. Calculate the amount each student saves during Week 5.

7. Calculate the amount each student saves during Week 10.

8. Create a table of weekly savings for the first ten weeks.

9. Graph the two sequences on the same coordinate plane.

10. Calculate the total savings for Student A over ten weeks.

11. Calculate the total savings for Student B over ten weeks.

12. Determine which student has saved more after ten weeks.

13. Explain why the geometric plan may eventually produce much larger weekly savings.

14. Identify the first week in which Student B's weekly savings exceed Student A's.

15. Discuss whether both savings plans are realistic over a long period.

16. Explain the difference between the weekly savings and total savings.

17. Predict how changing Student B's growth rate to 10% would affect the results.

18. Predict how changing Student A's weekly increase to $20 would affect the results.

19. Describe which plan you would prefer for a short savings period and explain your reasoning.

20. Explain how geometric sequences and series help people make financial decisions.