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.