1. What Is a Sequence?

2. Extending Numerical Patterns

Extending Numerical Patterns

A numerical pattern is an ordered list of numbers that follows a rule. When the numbers are arranged in order, the pattern is often called a sequence.

For example:

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

Each number in the sequence is called a term.

The rule for this sequence is:

Add 3 each time.

Understanding the rule allows us to extend the pattern, predict later terms, and describe how the numbers change.

 

 

 

6

Looking for the Rule

When you are given a numerical pattern, begin by comparing neighbouring terms.

Consider:

5, 9, 13, 17, 21, ...

Calculate the changes:

9 − 5 = 4

13 − 9 = 4

17 − 13 = 4

21 − 17 = 4

The same number is added each time.

Therefore, the rule is:

Start at 5 and add 4 each time.

The next terms are:

25, 29, 33, ...

Increasing Patterns

An increasing pattern produces larger values as the sequence continues.

For example:

12, 17, 22, 27, 32, ...

The rule is:

Add 5.

Therefore:

32 + 5 = 37

37 + 5 = 42

The next two terms are:

37, 42

Decreasing Patterns

Patterns can also decrease.

Consider:

50, 44, 38, 32, 26, ...

The change is:

−6

The rule can be described as:

Start at 50 and subtract 6 each time.

The next terms are:

26 − 6 = 20

20 − 6 = 14

So:

20, 14

Using a Difference Table

A difference table can make a pattern easier to see.

Consider:

7, 12, 17, 22, 27

Term number	Term	Change
1	7	—
2	12	+5
3	17	+5
4	22	+5
5	27	+5

Because the difference is always +5, the pattern has a constant difference.

A sequence with a constant difference is called an arithmetic sequence.

Multiplication Patterns

Not every pattern involves addition or subtraction.

Consider:

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

Compare consecutive terms:

6 ÷ 3 = 2

12 ÷ 6 = 2

24 ÷ 12 = 2

48 ÷ 24 = 2

The rule is:

Multiply by 2 each time.

The next terms are:

48 × 2 = 96

96 × 2 = 192

So:

96, 192

 

 

 

5

Division Patterns

A pattern can also involve repeated division.

For example:

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

Each term is divided by 2.

The next terms are:

10 ÷ 2 = 5

5 ÷ 2 = 2.5

Therefore:

5, 2.5

Describing a Pattern Clearly

A good pattern rule should include enough information for another person to recreate the sequence.

For:

8, 13, 18, 23, ...

saying:

"Add 5"

describes how the pattern changes.

An even clearer description is:

"Start at 8 and add 5 to get each new term."

This tells us both the starting value and the operation.

Recursive Rules

A rule that tells us how to find a term from the previous term is called a recursive rule.

For example:

10, 14, 18, 22, ...

The recursive rule is:

Start with 10 and add 4 to the previous term.

We can write:

next term = previous term + 4

Recursive rules are especially useful when extending a pattern one term at a time.

Predicting Terms Farther Along

Extending a sequence one term at a time works well when we only need the next few terms.

But suppose we have:

5, 8, 11, 14, 17, ...

and want the 50th term.

Adding 3 repeatedly 45 more times would be inefficient.

Instead, we can find a rule connecting the term number directly to the term value.

Term Numbers

Every term has a position.

For the sequence:

5, 8, 11, 14, 17, ...

Term number	Term value
1	5
2	8
3	11
4	14
5	17

The term number is usually represented by n.

We want an equation that produces the correct value for any n.

Finding a General Rule

The sequence:

5, 8, 11, 14, 17, ...

increases by 3.

Start with:

3n

This produces:

3, 6, 9, 12, 15, ...

But our sequence is:

5, 8, 11, 14, 17, ...

Each required value is 2 greater.

Therefore:

term = 3n + 2

We can write:

Tₙ = 3n + 2

where Tₙ means the value of term number n.

Check:

For n = 1:

T₁ = 3(1) + 2 = 5

For n = 4:

T₄ = 3(4) + 2 = 14

The rule works.

Predicting a Farther Term

Using:

Tₙ = 3n + 2

find the 50th term.

Substitute:

n = 50

T₅₀ = 3(50) + 2

T₅₀ = 150 + 2

T₅₀ = 152

We can find the 50th term without writing all 50 terms.

Another Example

Consider:

11, 18, 25, 32, 39, ...

The constant difference is:

+7

Start with:

7n

This gives:

7, 14, 21, 28, 35, ...

Compare with the actual sequence:

11, 18, 25, 32, 39, ...

Each value is 4 greater.

Therefore:

Tₙ = 7n + 4

Find the 20th term:

T₂₀ = 7(20) + 4

T₂₀ = 140 + 4

T₂₀ = 144

Arithmetic Sequence Formula

For an arithmetic sequence, we can also use:

Tₙ = a + (n − 1)d

where:

• Tₙ = nth term
• a = first term
• n = term number
• d = common difference

Example

Consider:

6, 10, 14, 18, ...

Here:

a = 6

d = 4

Therefore:

Tₙ = 6 + (n − 1)(4)

To find the 25th term:

T₂₅ = 6 + (25 − 1)(4)

T₂₅ = 6 + 96

T₂₅ = 102

Patterns with Negative Numbers

Patterns can cross zero and continue into negative numbers.

For example:

15, 10, 5, 0, −5, −10, ...

The rule is:

Subtract 5 each time.

The next terms are:

−15, −20

A negative value does not mean the pattern has ended.

Patterns with Decimals

The same strategies work with decimals.

Consider:

1.5, 2.0, 2.5, 3.0, 3.5, ...

The constant difference is:

+0.5

The next terms are:

4.0, 4.5

Patterns with Fractions

Consider:

1/2, 1, 3/2, 2, 5/2, ...

Each term increases by:

1/2

The next terms are:

3, 7/2

The numbers may look different, but the same pattern strategies apply.

Patterns with Changing Differences

Some sequences do not have a constant difference.

Consider:

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

Calculate the differences:

4 − 1 = 3

9 − 4 = 5

16 − 9 = 7

25 − 16 = 9

The differences are:

3, 5, 7, 9, ...

The differences themselves form a pattern.

The original sequence consists of square numbers:

1², 2², 3², 4², 5², ...

Therefore:

Tₙ = n²

The next term is:

6² = 36

 

 

 

5

Another Changing Pattern

Consider:

2, 5, 10, 17, 26, ...

Find the differences:

5 − 2 = 3

10 − 5 = 5

17 − 10 = 7

26 − 17 = 9

Again, the differences are:

3, 5, 7, 9, ...

Compare the terms with square numbers:

n²:

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

Our sequence:

2, 5, 10, 17, 26, ...

Each term is one greater.

Therefore:

Tₙ = n² + 1

The next term is:

6² + 1 = 37

Alternating Patterns

Some sequences alternate between different operations or values.

For example:

3, 8, 4, 9, 5, 10, 6, ...

One way to understand this is to separate the sequence into two patterns.

Odd-position terms:

3, 4, 5, 6, ...

Even-position terms:

8, 9, 10, ...

Both increase by 1.

Alternating patterns often require looking at the odd and even terms separately.

Multi-Step Rules

Some patterns use more than one operation.

Consider:

2, 5, 11, 23, 47, ...

Look at how each term changes:

2 × 2 + 1 = 5

5 × 2 + 1 = 11

11 × 2 + 1 = 23

23 × 2 + 1 = 47

The rule is:

Multiply by 2, then add 1.

The next term is:

47 × 2 + 1 = 95

Finding Missing Terms

Sometimes a sequence contains missing values.

For example:

12, __, 22, 27, 32

The visible values suggest a constant difference of +5.

Therefore:

12 + 5 = 17

So:

12, 17, 22, 27, 32

Another Example

3, 9, __, 81, 243

Each term is multiplied by 3.

Therefore:

9 × 3 = 27

The missing term is 27.

Creating Your Own Numerical Pattern

Creating a pattern requires choosing:

1. a starting value
2. a rule
3. several terms that follow the rule

Example

Starting value:

7

Rule:

Multiply by 2 and subtract 1

Generate the sequence:

7

7 × 2 − 1 = 13

13 × 2 − 1 = 25

25 × 2 − 1 = 49

49 × 2 − 1 = 97

So the pattern is:

7, 13, 25, 49, 97, ...

A clear description would be:

Start at 7. Multiply each term by 2 and subtract 1 to obtain the next term.

More Than One Possible Pattern

A short list of numbers can sometimes fit more than one rule.

Consider:

2, 4, 6, ...

The most obvious rule is:

Add 2 each time.

This predicts:

8, 10, ...

But with only three terms, more complicated rules could also be invented.

For school pattern problems, we normally look for the simplest reasonable rule that fits all the given terms.

The more terms we are given, the more evidence we have for the intended pattern.

Real-World Connections

Numerical patterns appear in many situations.

Saving Money

If you save $25 each week, your total savings might follow:

25, 50, 75, 100, 125, ...

Seating

If each new row contains 4 more seats than the previous row:

20, 24, 28, 32, ...

Population Growth

If a population doubles during each time interval:

100, 200, 400, 800, ...

Depreciation

If an object's value is repeatedly multiplied by 0.8:

1000, 800, 640, 512, ...

Recognizing the pattern allows us to make predictions.

A Strategy for Investigating Patterns

When you encounter a new sequence, ask:

Are the terms increasing or decreasing?

Then check:

Is the same amount being added or subtracted?

If not:

Is each term multiplied or divided by the same number?

If not:

Do the differences themselves form a pattern?

Then consider:

• square numbers
• cube numbers
• alternating patterns
• repeated multi-step rules
• relationships between term number and term value

Finally, test your proposed rule against every term given.

Common Mistakes

Looking only at the first two terms

A rule should fit all the terms provided.

Confusing addition with multiplication

For:

5, 10, 20, 40, ...

the difference is not constant. The terms are multiplied by 2.

Forgetting the starting value

"Add 4" does not completely define a sequence unless the starting term is known.

Assuming every pattern has a constant difference

Some patterns involve multiplication, changing differences, squares, or alternating rules.

Making an arithmetic error when extending a correct pattern

After finding the rule, apply it carefully to each new term.

Extending a long sequence one term at a time

For distant terms in arithmetic sequences, an nth-term rule is much more efficient.

Key Terms

Numerical pattern — an ordered collection of numbers generated according to a rule.

Sequence — an ordered list of terms.

Term — an individual number in a sequence.

Term number — the position of a term within a sequence.

Pattern rule — the instruction or relationship used to generate the terms.

Recursive rule — a rule that uses one term to determine the next term.

Arithmetic sequence — a sequence with a constant difference between consecutive terms.

Common difference — the constant amount added or subtracted in an arithmetic sequence.

Geometric pattern — a pattern in which terms are repeatedly multiplied by the same factor.

General rule — an equation connecting the term number to the term value.

Nth term — a general expression that gives the value of any term based on its position n.

Alternating pattern — a sequence in which two or more patterns alternate.

Key Takeaways

• Numerical patterns follow identifiable rules.
• Each number in a sequence is called a term.
• Comparing consecutive terms can reveal how a pattern changes.
• Constant addition or subtraction produces an arithmetic sequence.
• Some patterns involve repeated multiplication or division.
• Some patterns have changing differences or multi-step rules.
• Recursive rules describe how to generate the next term.
• An nth-term rule allows us to predict terms much farther along a sequence.
• For an arithmetic sequence, Tₙ = a + (n − 1)d.
• A proposed rule should fit all the given terms.
• Numerical patterns can involve positive numbers, negative numbers, fractions, and decimals.
• Creating your own pattern requires both a starting value and a clear rule.

Check Your Understanding

1. Find the next three terms:

   6, 10, 14, 18, ...

2. Describe the rule for:

   45, 38, 31, 24, ...

3. Find the next two terms:

   5, 15, 45, 135, ...

4. Find the missing term:

   18, 24, __, 36, 42

5. Find the next two terms:

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

6. Describe the rule for:

   4, 9, 19, 39, 79, ...

7. The sequence is:

   8, 13, 18, 23, ...

   Find the 20th term.

8. Find an nth-term rule for:

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

9. Use your rule from Question 8 to find the 50th term.
10. Determine the next term:

3, 6, 4, 8, 5, 10, 6, ...

11. A sequence begins at 12. Each new term is found by multiplying the previous term by 2 and subtracting 3. Write the first six terms.
12. Create your own numerical pattern containing at least six terms. State the starting value, describe the rule clearly, and predict the tenth term.

 

 

 

 

3. Recursive Rules

Recursive Rules

A recursive rule describes a sequence by explaining how each new term is generated from one or more terms that come before it.

For example:

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

A recursive description is:

Start at 4 and add 3 each time.

The starting value is important. Saying only "add 3" does not tell us where the sequence begins.

We can write the rule mathematically as:

a₁ = 4

aₙ = aₙ₋₁ + 3, for n ≥ 2

This means:

• a₁ is the first term
• aₙ represents the current term
• aₙ₋₁ represents the previous term
• each term is 3 greater than the previous term

 

 

 

5

What Makes a Rule Recursive?

The word recursive means that the rule refers back to something that came before.

Consider:

10, 15, 20, 25, 30, ...

To generate the sequence:

10 → add 5 → 15

15 → add 5 → 20

20 → add 5 → 25

25 → add 5 → 30

Each new term depends on the previous term.

The recursive rule is:

a₁ = 10

aₙ = aₙ₋₁ + 5

The first equation gives the starting value.

The second equation tells us how to generate the remaining terms.

Understanding the Notation

Recursive notation may look complicated at first, but each symbol has a simple meaning.

a₁

means the first term.

a₂

means the second term.

a₃

means the third term.

More generally:

aₙ

means the nth term.

The expression:

aₙ₋₁

means the term immediately before aₙ.

For example, if n = 6:

aₙ = a₆

and:

aₙ₋₁ = a₅

So the rule:

aₙ = aₙ₋₁ + 4

means:

Take the previous term and add 4.

Generating a Sequence from a Recursive Rule

Suppose:

a₁ = 6

aₙ = aₙ₋₁ + 4

Start with:

a₁ = 6

Now generate each term.

a₂ = 6 + 4 = 10

a₃ = 10 + 4 = 14

a₄ = 14 + 4 = 18

a₅ = 18 + 4 = 22

Therefore, the sequence is:

6, 10, 14, 18, 22, ...

Recursive Rules for Decreasing Sequences

Recursive sequences can decrease as well as increase.

Consider:

30, 26, 22, 18, 14, ...

Each term is 4 less than the previous term.

Therefore:

a₁ = 30

aₙ = aₙ₋₁ − 4

Check:

30 − 4 = 26

26 − 4 = 22

22 − 4 = 18

The rule correctly generates the sequence.

Finding the Starting Value

A recursive rule is incomplete without enough information to begin the sequence.

Suppose we are given:

aₙ = aₙ₋₁ + 7

Can we determine the sequence?

No.

It could be:

1, 8, 15, 22, ...

or:

10, 17, 24, 31, ...

or:

−5, 2, 9, 16, ...

All these sequences follow the rule add 7.

We also need a starting value.

For example:

a₁ = 3

Now the sequence is completely defined:

3, 10, 17, 24, 31, ...

Writing a Recursive Rule from a Sequence

Suppose we are given:

8, 13, 18, 23, 28, ...

First identify the starting value:

a₁ = 8

Next determine how the terms change:

13 − 8 = 5

18 − 13 = 5

23 − 18 = 5

The sequence increases by 5 each time.

Therefore:

a₁ = 8

aₙ = aₙ₋₁ + 5

Another Example

Write a recursive rule for:

50, 42, 34, 26, 18, ...

Starting value:

a₁ = 50

Difference:

42 − 50 = −8

34 − 42 = −8

Therefore:

a₁ = 50

aₙ = aₙ₋₁ − 8

Recursive Multiplication Rules

Recursive rules are not limited to addition and subtraction.

Consider:

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

Each term is twice the previous term.

Therefore:

a₁ = 3

aₙ = 2aₙ₋₁

The sequence develops as:

3 → 6 → 12 → 24 → 48

This is a recursive rule involving multiplication.

 

 

 

4

Recursive Division Rules

Consider:

128, 64, 32, 16, 8, ...

Each term is half the previous term.

We could write:

a₁ = 128

aₙ = aₙ₋₁ ÷ 2

or equivalently:

aₙ = 0.5aₙ₋₁

Generate the next term:

8 ÷ 2 = 4

Then:

4 ÷ 2 = 2

So the sequence continues:

4, 2, ...

Multi-Step Recursive Rules

Some recursive rules involve more than one operation.

Consider:

2, 5, 11, 23, 47, ...

Compare consecutive terms:

2 × 2 + 1 = 5

5 × 2 + 1 = 11

11 × 2 + 1 = 23

23 × 2 + 1 = 47

Therefore:

a₁ = 2

aₙ = 2aₙ₋₁ + 1

To find the next term:

a₆ = 2(47) + 1

a₆ = 95

Another Multi-Step Example

Suppose:

a₁ = 10

aₙ = 3aₙ₋₁ − 2

Generate the first five terms.

First:

a₁ = 10

Then:

a₂ = 3(10) − 2 = 28

a₃ = 3(28) − 2 = 82

a₄ = 3(82) − 2 = 244

a₅ = 3(244) − 2 = 730

Therefore:

10, 28, 82, 244, 730, ...

When a recursive rule contains multiple operations, remember to follow the correct order of operations.

Recursive Rules with Negative Numbers

Suppose:

a₁ = 12

aₙ = aₙ₋₁ − 5

Generate several terms:

12

12 − 5 = 7

7 − 5 = 2

2 − 5 = −3

−3 − 5 = −8

Therefore:

12, 7, 2, −3, −8, ...

A sequence can continue through zero into negative values.

Recursive Rules with Decimals

Suppose:

a₁ = 20

aₙ = 0.5aₙ₋₁

Then:

a₁ = 20

a₂ = 10

a₃ = 5

a₄ = 2.5

a₅ = 1.25

Therefore:

20, 10, 5, 2.5, 1.25, ...

Recursive rules work with decimals and fractions just as they do with whole numbers.

Pattern Descriptions and Recursive Rules

A pattern can often be described using words or mathematical notation.

Consider:

7, 11, 15, 19, 23, ...

Word Description

Start at 7 and add 4 each time.

Recursive Rule

a₁ = 7

aₙ = aₙ₋₁ + 4

These descriptions communicate the same information.

The recursive notation is simply a more formal mathematical way of describing the pattern.

Translating Words into Recursive Rules

Learning to move between verbal descriptions and symbolic rules is an important mathematical skill.

Recursive Rules vs. Explicit Rules

There are two common ways to describe sequences.

Recursive Rule

A recursive rule uses previous terms.

For:

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

we can write:

a₁ = 4

aₙ = aₙ₋₁ + 3

To find the fifth term, we normally generate the earlier terms first.

Explicit Rule

An explicit rule gives a term directly from its position.

For the same sequence:

aₙ = 3n + 1

To find the 100th term:

a₁₀₀ = 3(100) + 1

a₁₀₀ = 301

We do not need to calculate the first 99 terms.

Comparing Recursive and Explicit Rules

Neither type is automatically "better." They describe a sequence in different ways.

When Recursive Rules Are Useful

Recursive rules are especially useful when:

• each stage depends naturally on the previous stage
• we want to generate consecutive values
• the relationship between neighbouring terms is simple
• a process repeats over time

For example, suppose a bank account contains $1,000 and increases by 5% each year.

A recursive model could be:

a₁ = 1000

aₙ = 1.05aₙ₋₁

Each year's balance is calculated from the previous year's balance.

Real-World Example: Population

Suppose a bacteria population begins at 200 and doubles every hour.

A recursive rule is:

a₁ = 200

aₙ = 2aₙ₋₁

The sequence is:

200, 400, 800, 1600, 3200, ...

The recursive rule represents the repeated growth process naturally.

Real-World Example: Depreciation

Suppose a computer is initially worth $2,000 and retains 80% of its value each year.

A recursive model is:

a₁ = 2000

aₙ = 0.8aₙ₋₁

Its values are:

2000

1600

1280

1024

819.20

Each year's value depends directly on the previous year's value.

 

 

 

5

Rules Using More Than One Previous Term

Some sequences depend on more than one previous term.

A famous example is the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, ...

Starting values:

a₁ = 1

a₂ = 1

Each new term is the sum of the previous two terms:

aₙ = aₙ₋₁ + aₙ₋₂

For example:

a₆ = a₅ + a₄

a₆ = 5 + 3

a₆ = 8

This sequence needs two starting values because each new term depends on two earlier terms.

 

 

 

5

Checking a Recursive Rule

Suppose someone claims that:

a₁ = 5

aₙ = aₙ₋₁ + 6

generates:

5, 11, 17, 24, 29, ...

Check each step.

5 + 6 = 11 ✓

11 + 6 = 17 ✓

17 + 6 = 23, not 24 ✗

Therefore, the sequence does not follow the stated recursive rule.

A rule must correctly generate every term.

Creating Your Own Recursive Sequence

To create a recursive sequence:

1. Choose a starting value.
2. Choose an operation or combination of operations.
3. Write the recursive rule.
4. Generate several terms.
5. Check that every term follows the rule.

Example

Choose:

a₁ = 4

Rule:

aₙ = 2aₙ₋₁ − 3

Generate:

a₁ = 4

a₂ = 2(4) − 3 = 5

a₃ = 2(5) − 3 = 7

a₄ = 2(7) − 3 = 11

a₅ = 2(11) − 3 = 19

Therefore:

4, 5, 7, 11, 19, ...

A Strategy for Writing Recursive Rules

When given a sequence:

First: identify the first term.

Write:

a₁ = ...

Next: compare consecutive terms.

Ask:

• Is the same amount added?
• Is the same amount subtracted?
• Is each term multiplied by the same number?
• Is each term divided by the same number?
• Is there a combination of operations?

Then: write the relationship using aₙ and aₙ₋₁.

Finally, test your rule using several terms.

Common Mistakes

Forgetting the starting value

The rule:

aₙ = aₙ₋₁ + 5

does not uniquely define a sequence without a starting value.

Confusing aₙ and aₙ₋₁

aₙ represents the current term.

aₙ₋₁ represents the previous term.

Using n instead of the previous term

The rule:

aₙ = 3n + 2

is an explicit rule, not a recursive rule.

Looking only at the first two terms

Always test the proposed rule against several terms.

Forgetting order of operations

For:

aₙ = 2aₙ₋₁ + 3

multiply the previous term by 2 before adding 3.

Assuming every recursive rule uses one previous term

Some sequences, such as Fibonacci, depend on two or more earlier terms.

Key Terms

Recursive rule — a rule that defines terms of a sequence using one or more previous terms.

Sequence — an ordered list of numbers or other mathematical objects.

Term — an individual value in a sequence.

Starting value — the initial term needed to begin generating a recursive sequence.

Previous term — the term immediately before the current term.

aₙ — notation representing the nth term of a sequence.

aₙ₋₁ — notation representing the term immediately before the nth term.

aₙ₋₂ — notation representing the term two positions before the nth term.

Recursive sequence — a sequence generated using a recursive rule.

Explicit rule — a rule that calculates a term directly from its position rather than from previous terms.

Arithmetic sequence — a sequence with a constant difference between consecutive terms.

Geometric sequence — a sequence with a constant multiplication factor between consecutive terms.

Key Takeaways

• A recursive rule defines each new term using one or more earlier terms.
• A recursive sequence requires enough starting values to begin the process.
• The notation aₙ₋₁ means the term immediately before aₙ.
• Recursive rules can involve addition, subtraction, multiplication, division, or combinations of operations.
• A verbal pattern description can usually be translated into recursive notation.
• For 4, 7, 10, 13, ..., the recursive rule is a₁ = 4, aₙ = aₙ₋₁ + 3.
• Recursive rules show clearly how a pattern changes from one term to the next.
• Explicit rules allow us to calculate distant terms directly.
• Some recursive sequences depend on more than one previous term.
• A recursive rule should always be tested against the given sequence.

Check Your Understanding

1. Explain what makes a rule recursive.

2. Generate the first five terms:

   a₁ = 3

   aₙ = aₙ₋₁ + 6

3. Generate the first five terms:

   a₁ = 40

   aₙ = aₙ₋₁ − 7

4. Generate the first six terms:

   a₁ = 2

   aₙ = 3aₙ₋₁

5. Write a recursive rule for:

   6, 10, 14, 18, 22, ...

6. Write a recursive rule for:

   100, 90, 80, 70, 60, ...

7. Write a recursive rule for:

   5, 15, 45, 135, ...

8. Write a recursive rule for:

   3, 7, 15, 31, 63, ...

9. Explain why aₙ = aₙ₋₁ + 4 is not enough information to determine a unique sequence.
10. A student describes the sequence 12, 17, 22, 27, ... as "start at 12 and add 5." Write this description as a recursive rule.
11. Compare the recursive rule a₁ = 2, aₙ = aₙ₋₁ + 3 with the explicit rule aₙ = 3n − 1. Show that both generate the same first five terms.
12. Create your own recursive rule involving at least two mathematical operations. Generate the first six terms and describe the rule in words.

4. Explicit Rules

Explicit Rules

An explicit rule is a formula that allows us to calculate any term in a sequence directly from its position.

Unlike a recursive rule, we do not need to know the previous term.

For example, consider:

5, 8, 11, 14, 17, ...

An explicit rule for this sequence is:

aₙ = 3n + 2

where:

• aₙ = value of the nth term
• n = position or term number

If we want the 20th term, we substitute n = 20:

a₂₀ = 3(20) + 2

a₂₀ = 60 + 2

a₂₀ = 62

We can find the 20th term without calculating the first 19 terms.

 

 

 

5

What Makes a Rule Explicit?

An explicit rule connects the term number directly to the term value.

Consider:

4, 9, 14, 19, 24, ...

An explicit rule is:

aₙ = 5n − 1

To find a term, simply substitute its position.

For the 6th term:

a₆ = 5(6) − 1

a₆ = 30 − 1

a₆ = 29

For the 100th term:

a₁₀₀ = 5(100) − 1

a₁₀₀ = 500 − 1

a₁₀₀ = 499

This is the main advantage of an explicit rule: distant terms can be calculated immediately.

Understanding the Notation

In:

aₙ = 4n + 3

the symbol n represents the term number.

The symbol aₙ represents the value at that position.

For example:

n	Meaning
1	first term
2	second term
5	fifth term
20	twentieth term
100	hundredth term

If:

aₙ = 4n + 3

then:

a₁ = 4(1) + 3 = 7

a₂ = 4(2) + 3 = 11

a₃ = 4(3) + 3 = 15

So the sequence is:

7, 11, 15, 19, ...

Generating a Sequence from an Explicit Rule

Suppose:

aₙ = 6n − 2

Generate the first five terms.

For n = 1:

a₁ = 6(1) − 2 = 4

For n = 2:

a₂ = 6(2) − 2 = 10

For n = 3:

a₃ = 6(3) − 2 = 16

For n = 4:

a₄ = 6(4) − 2 = 22

For n = 5:

a₅ = 6(5) − 2 = 28

Therefore:

4, 10, 16, 22, 28, ...

Substitution

Using explicit rules requires careful substitution.

Suppose:

aₙ = 7n + 4

Find a₁₂.

Because we want the 12th term:

n = 12

Substitute:

a₁₂ = 7(12) + 4

a₁₂ = 84 + 4

a₁₂ = 88

A useful habit is to write the substitution step before calculating.

Finding Distant Terms

Consider:

aₙ = 8n − 5

Find the 200th term.

Substitute:

n = 200

a₂₀₀ = 8(200) − 5

a₂₀₀ = 1600 − 5

a₂₀₀ = 1595

We did not need to calculate terms 1 through 199.

This makes explicit rules particularly useful when dealing with large term numbers.

Explicit Rules for Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

For example:

7, 12, 17, 22, 27, ...

The common difference is:

d = 5

For an arithmetic sequence, the explicit rule can be written:

aₙ = a₁ + (n − 1)d

where:

• aₙ = nth term
• a₁ = first term
• n = term number
• d = common difference

Using the Arithmetic Sequence Formula

For:

7, 12, 17, 22, ...

we have:

a₁ = 7

d = 5

Therefore:

aₙ = 7 + (n − 1)(5)

Simplify:

aₙ = 7 + 5n − 5

aₙ = 5n + 2

Now any term can be calculated directly.

For example:

a₅₀ = 5(50) + 2

a₅₀ = 252

 

 

 

5

Finding an Explicit Rule from a Sequence

Consider:

8, 11, 14, 17, 20, ...

First identify the common difference:

d = 3

Start with:

3n

This produces:

3, 6, 9, 12, 15, ...

Compare this with the actual sequence:

8, 11, 14, 17, 20, ...

Each actual term is 5 greater.

Therefore:

aₙ = 3n + 5

Check:

a₁ = 3(1) + 5 = 8 ✓

a₂ = 3(2) + 5 = 11 ✓

a₃ = 3(3) + 5 = 14 ✓

Another Example

Find an explicit rule for:

14, 20, 26, 32, 38, ...

Common difference:

d = 6

Start with:

6n

This gives:

6, 12, 18, 24, 30, ...

The actual sequence is always 8 greater.

Therefore:

aₙ = 6n + 8

Check the first term:

a₁ = 6(1) + 8 = 14

The rule works.

Explicit Rules for Decreasing Sequences

Explicit rules can also describe decreasing patterns.

Consider:

30, 26, 22, 18, 14, ...

The common difference is:

−4

Use:

aₙ = a₁ + (n − 1)d

Substitute:

aₙ = 30 + (n − 1)(−4)

Simplify:

aₙ = 30 − 4n + 4

aₙ = 34 − 4n

Check:

a₁ = 34 − 4(1) = 30

a₂ = 34 − 4(2) = 26

The rule works.

Finding the Term Number

Explicit rules can also be used in reverse.

Suppose:

aₙ = 5n + 2

Which term has a value of 72?

Set:

aₙ = 72

Therefore:

72 = 5n + 2

Subtract 2:

70 = 5n

Divide by 5:

n = 14

Therefore, 72 is the 14th term.

Is a Number in the Sequence?

Suppose:

aₙ = 4n + 1

Is 101 in the sequence?

Set:

101 = 4n + 1

100 = 4n

n = 25

Since 25 is a positive whole-number term position:

Yes, 101 is the 25th term.

Now ask whether 100 is in the sequence:

100 = 4n + 1

99 = 4n

n = 24.75

A term number must be a positive whole number.

Therefore:

100 is not a term in the sequence.

Recursive Rules

Recall that a recursive rule defines each term using the previous term.

For:

5, 8, 11, 14, 17, ...

a recursive rule is:

a₁ = 5

aₙ = aₙ₋₁ + 3

To generate later terms, we repeatedly add 3.

Explicit Rule for the Same Sequence

The same sequence can be written explicitly as:

aₙ = 3n + 2

Now we can calculate any term directly.

For example:

a₁₀₀ = 3(100) + 2

a₁₀₀ = 302

This is much faster than applying the recursive rule 99 times.

Comparing Explicit and Recursive Rules

Consider:

10, 14, 18, 22, 26, ...

Recursive

a₁ = 10

aₙ = aₙ₋₁ + 4

This tells us:

Start at 10 and repeatedly add 4.

Explicit

aₙ = 4n + 6

This tells us:

Multiply the term number by 4 and add 6.

Both rules generate exactly the same sequence.

Recursive Rule	Explicit Rule
Uses previous term(s)	Uses term number n
Requires starting value	Calculates term directly
Shows how sequence changes	Shows position-value relationship
Useful for generating consecutive terms	Useful for distant terms

Why Both Forms Are Useful

Recursive rules are often easier to identify because we naturally look at how consecutive terms change.

For example:

20, 27, 34, 41, ...

It is easy to see:

add 7

So:

a₁ = 20

aₙ = aₙ₋₁ + 7

But if we want the 500th term, an explicit rule is much more efficient:

aₙ = 7n + 13

Then:

a₅₀₀ = 7(500) + 13

a₅₀₀ = 3513

Converting a Recursive Arithmetic Rule to Explicit Form

Suppose:

a₁ = 9

aₙ = aₙ₋₁ + 4

We know:

a₁ = 9

d = 4

Use:

aₙ = a₁ + (n − 1)d

Therefore:

aₙ = 9 + 4(n − 1)

aₙ = 9 + 4n − 4

aₙ = 4n + 5

So the recursive and explicit forms describe the same sequence.

Explicit Rules Beyond Arithmetic Sequences

Not every explicit rule is linear.

Consider:

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

These are square numbers.

The explicit rule is:

aₙ = n²

Therefore:

a₁₀ = 10² = 100

a₅₀ = 50² = 2500

Another example:

2, 5, 10, 17, 26, ...

This follows:

aₙ = n² + 1

because:

1² + 1 = 2

2² + 1 = 5

3² + 1 = 10

4² + 1 = 17

 

 

 

6

Explicit Rules for Geometric Sequences

A sequence that repeatedly multiplies by the same factor is called a geometric sequence.

For example:

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

The recursive rule is:

a₁ = 3

aₙ = 2aₙ₋₁

An explicit rule is:

aₙ = 3(2)ⁿ⁻¹

Check the fourth term:

a₄ = 3(2)⁴⁻¹

a₄ = 3(2)³

a₄ = 3(8)

a₄ = 24

General Geometric Formula

For a geometric sequence:

aₙ = a₁rⁿ⁻¹

where:

• a₁ = first term
• r = common ratio
• n = term number

Example

Consider:

5, 15, 45, 135, ...

Here:

a₁ = 5

r = 3

Therefore:

aₙ = 5(3)ⁿ⁻¹

Find the 8th term:

a₈ = 5(3)⁷

a₈ = 5(2187)

a₈ = 10,935

The explicit formula allows us to jump directly to the eighth term.

Real-World Example: Saving Money

Suppose you begin with $50 and add $20 each week.

The amounts are:

50, 70, 90, 110, 130, ...

An explicit rule is:

aₙ = 50 + 20(n − 1)

or:

aₙ = 20n + 30

After 30 weeks:

a₃₀ = 20(30) + 30

a₃₀ = $630

Real-World Example: Doubling

Suppose a bacteria population begins with 100 bacteria and doubles during each time interval.

The sequence is:

100, 200, 400, 800, 1600, ...

An explicit rule is:

aₙ = 100(2)ⁿ⁻¹

This model allows us to calculate any stage directly.

 

 

 

5

Choosing Between Explicit and Recursive Rules

Use a recursive rule when you want to emphasize:

• how one term changes into the next
• a repeated process
• consecutive terms

Use an explicit rule when you want to:

• find a distant term
• connect term position directly to value
• avoid calculating every previous term
• determine whether a particular value appears in a sequence

In many situations, both forms are useful.

A Strategy for Using Explicit Rules

When given an explicit formula:

First: identify the requested term number.

Next: set n equal to that term number.

Then: substitute n into the formula.

Next: follow the order of operations carefully.

Finally: check whether the result is reasonable for the sequence.

For example:

aₙ = 6n − 1

Find a₂₅.

n = 25

a₂₅ = 6(25) − 1

a₂₅ = 150 − 1

a₂₅ = 149

Common Mistakes

Using the term value instead of the term number

If you are finding the 20th term, substitute:

n = 20

Starting with n = 0 when the sequence begins at n = 1

Unless otherwise stated, sequences in these notes begin with:

n = 1

Confusing explicit and recursive rules

aₙ = 4n + 3 is explicit.

aₙ = aₙ₋₁ + 4 is recursive.

Forgetting the (n − 1) in arithmetic formulas

The arithmetic formula is:

aₙ = a₁ + (n − 1)d

not:

aₙ = a₁ + nd

Forgetting the (n − 1) exponent in geometric formulas

The standard geometric formula is:

aₙ = a₁rⁿ⁻¹

because the first term has undergone zero multiplications by r.

Listing every previous term unnecessarily

If an explicit rule is available, use it to calculate the required term directly.

Key Terms

Explicit rule — a formula that calculates a term directly from its position.

Sequence — an ordered list of terms.

Term — an individual value in a sequence.

Term number — the position of a term in a sequence.

nth term — the term at position n.

aₙ — notation representing the nth term.

Substitution — replacing a variable with a known value.

Arithmetic sequence — a sequence with a constant difference between consecutive terms.

Common difference (d) — the constant amount added or subtracted in an arithmetic sequence.

Geometric sequence — a sequence in which each term is obtained by multiplying the previous term by a constant factor.

Common ratio (r) — the constant multiplication factor in a geometric sequence.

Recursive rule — a rule that defines terms using one or more previous terms.

Key Takeaways

• An explicit rule calculates a term directly from its term number.
• The variable n represents the position of a term.
• aₙ represents the value of the nth term.
• Explicit rules allow distant terms to be calculated without listing earlier terms.
• For arithmetic sequences: aₙ = a₁ + (n − 1)d.
• For geometric sequences: aₙ = a₁rⁿ⁻¹.
• Explicit rules can also describe patterns such as aₙ = n².
• Recursive rules use previous terms; explicit rules use the term number.
• The same sequence can often be represented both recursively and explicitly.
• When using an explicit formula, identify n, substitute carefully, and follow the order of operations.
• Explicit formulas are especially useful for finding distant terms such as the 50th, 100th, or 1000th term.

Check Your Understanding

1. Explain what makes a rule explicit.
2. Given aₙ = 4n + 3, find a₁, a₂, a₃, and a₄.
3. Given aₙ = 7n − 2, find the 20th term.
4. Given aₙ = 5n + 8, find the 100th term.

5. Find an explicit rule for:

   6, 10, 14, 18, 22, ...

6. Find an explicit rule for:

   15, 22, 29, 36, 43, ...

7. Find an explicit rule for:

   40, 35, 30, 25, 20, ...

8. The sequence has the rule aₙ = 6n + 1. Determine whether 103 is a term in the sequence.

9. The recursive rule is:

   a₁ = 8

   aₙ = aₙ₋₁ + 5

   Write an explicit rule for the same sequence.

10. Compare the recursive rule a₁ = 4, aₙ = aₙ₋₁ + 3 with the explicit rule aₙ = 3n + 1. Explain how the two rules describe the same sequence.
11. A geometric sequence begins:

4, 12, 36, 108, ...

Write an explicit rule and use it to find the 8th term.

12. Explain why an explicit rule is generally more efficient than a recursive rule when finding the 500th term of a sequence.

 

 

 

5. Sequences in Nature and Society

Sequences in Nature and Society

Sequences are not only mathematical exercises. They appear whenever quantities change in a predictable way over time, position, or repeated stages.

Examples can be found in:

• population growth
• savings and investments
• loan payments
• plant growth
• animal populations
• patterns in flowers and seeds
• depreciation of objects
• disease spread
• construction and design
• computer science

By identifying a pattern and representing it as a sequence, we can describe what has already happened and make predictions about what might happen next.

 

 

 

5

From Real Situations to Sequences

Suppose someone saves $50 every month.

Their total savings might be:

50, 100, 150, 200, 250, ...

This sequence increases by a constant amount:

+50

We can describe it recursively:

a₁ = 50

aₙ = aₙ₋₁ + 50

We can also describe it explicitly:

aₙ = 50n

The sequence provides a simple mathematical model of the person's savings.

What Is a Mathematical Model?

A mathematical model uses mathematics to represent some aspect of a real situation.

Models can use:

• sequences
• equations
• graphs
• tables
• diagrams
• functions

A model usually simplifies reality.

For example, the savings sequence:

50, 100, 150, 200, ...

assumes that exactly $50 is saved every month.

Real life may be more complicated. Some months the person might save more or less.

The model is useful because it allows us to make predictions if the pattern continues.

Arithmetic Sequences in Everyday Life

An arithmetic sequence changes by the same amount during each step.

For example:

20, 25, 30, 35, 40, ...

The common difference is:

d = 5

Arithmetic sequences can model situations involving constant increases or decreases.

Examples include:

• saving the same amount each week
• adding the same number of seats to each row
• increasing a training distance by the same amount each week
• losing the same number of units from inventory each day

Example: Theatre Seating

Suppose the first row of a theatre contains 18 seats and each new row contains 3 more seats than the previous row.

The sequence is:

18, 21, 24, 27, 30, ...

The recursive rule is:

a₁ = 18

aₙ = aₙ₋₁ + 3

The explicit rule is:

aₙ = 18 + 3(n − 1)

which simplifies to:

aₙ = 3n + 15

To find the number of seats in row 25:

a₂₅ = 3(25) + 15

a₂₅ = 90

The 25th row contains 90 seats.

Population Growth

Sequences can be used to model population changes.

Suppose a bacteria population begins at 500 and doubles every hour.

The population sequence is:

500, 1000, 2000, 4000, 8000, ...

This is a geometric sequence because each term is obtained by multiplying the previous term by the same number.

The common ratio is:

r = 2

Recursive rule:

a₁ = 500

aₙ = 2aₙ₋₁

Explicit rule:

aₙ = 500(2)ⁿ⁻¹

 

 

 

5

Predicting a Population

Using:

aₙ = 500(2)ⁿ⁻¹

find the population at the sixth stage.

a₆ = 500(2)⁵

a₆ = 500(32)

a₆ = 16,000

If the model continues to apply, the population at the sixth stage will be 16,000 bacteria.

This demonstrates how sequences allow us to predict future outcomes.

Real Populations Have Limits

A simple geometric sequence can predict extremely rapid population growth.

But real populations cannot usually grow exponentially forever.

Eventually, factors such as:

• food availability
• space
• disease
• competition
• predators
• water
• environmental conditions

can slow growth.

So the sequence may be useful for short-term predictions but unrealistic over very long periods.

More advanced population models account for environmental limits and carrying capacity.

This highlights an important principle:

A mathematical pattern may be correct while its real-world application has limits.

Sequences in Finance

Finance provides many examples of sequences.

Suppose you deposit $100 into a savings account every month without earning interest.

Your deposits form an arithmetic pattern:

100, 200, 300, 400, 500, ...

The amount increases by:

$100 per month

However, if an existing balance grows by a percentage each year, the situation can form a geometric sequence.

Compound Growth

Suppose $1,000 is invested and increases by 5% each year.

Each year the balance is multiplied by:

1.05

The sequence begins approximately:

1000, 1050, 1102.50, 1157.63, 1215.51, ...

The recursive rule is:

a₁ = 1000

aₙ = 1.05aₙ₋₁

An explicit rule is:

aₙ = 1000(1.05)ⁿ⁻¹

This sequence can be used to predict future account balances.

 

 

 

5

Worked Example: Investment

Suppose an investment begins at $2,000 and increases by 4% each year.

The multiplier is:

1 + 0.04 = 1.04

The sequence can be modelled by:

aₙ = 2000(1.04)ⁿ⁻¹

Find the value at the tenth stage:

a₁₀ = 2000(1.04)⁹

a₁₀ ≈ 2846.62

The model predicts approximately $2,846.62.

Notice that the amount added each year is not constant. The percentage increase is constant.

Depreciation

Sequences can also model quantities that decrease.

Suppose a computer is worth $2,500 and retains 80% of its value each year.

The sequence begins:

2500, 2000, 1600, 1280, 1024, ...

Each term is multiplied by:

0.8

Recursive rule:

a₁ = 2500

aₙ = 0.8aₙ₋₁

Explicit rule:

aₙ = 2500(0.8)ⁿ⁻¹

This is a geometric sequence with:

r = 0.8

Because:

0 < r < 1

the sequence decreases.

Worked Example: Depreciation

Using:

aₙ = 2500(0.8)ⁿ⁻¹

find the value at the fifth stage.

a₅ = 2500(0.8)⁴

a₅ = 2500(0.4096)

a₅ = $1024

According to the model, the computer is worth $1,024.

Again, a real computer's resale value may not follow this exact pattern. The sequence is a model.

Fibonacci Sequence

One of the most famous sequences associated with patterns in nature is the Fibonacci sequence.

A common version begins:

1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Each new term is the sum of the previous two terms.

So:

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

The recursive rule is:

a₁ = 1

a₂ = 1

aₙ = aₙ₋₁ + aₙ₋₂

 

 

 

5

Fibonacci Patterns in Nature

Numbers related to the Fibonacci sequence can appear in biological growth patterns.

Examples often discussed include:

• spiral arrangements in sunflower seed heads
• spiral patterns in pinecones
• arrangements of leaves around stems
• numbers of petals in some flowers

These patterns can arise because biological growth processes often involve efficient arrangements of new structures around existing ones.

However, nature does not follow the Fibonacci sequence perfectly in every case.

Not every flower has a Fibonacci number of petals, and not every spiral found in nature is a Fibonacci spiral.

The sequence is a useful mathematical description of some natural patterns, not a universal law of nature.

Patterns in Plant Growth

Plants often produce structures according to repeated developmental rules.

Leaves may emerge at regular positions along stems.

Branches may form repeated arrangements.

Seeds may appear in spiral patterns.

These patterns can sometimes be represented mathematically using sequences.

 

 

 

6

Animal Populations

Sequences can also model animal populations.

Suppose a protected population contains 120 animals and increases by approximately 15 animals each year.

A simple arithmetic model is:

120, 135, 150, 165, 180, ...

The explicit rule is:

aₙ = 120 + 15(n − 1)

This model assumes a constant increase of 15 animals per year.

If the population instead increases by 10% each year, a geometric model may be more appropriate:

aₙ = 120(1.10)ⁿ⁻¹

The choice of sequence depends on how the population changes.

Arithmetic or Geometric?

A useful question is:

Does the quantity change by a constant amount or a constant factor?

Constant Amount

Example:

100, 120, 140, 160, ...

Difference:

+20

This is arithmetic.

Constant Factor

Example:

100, 120, 144, 172.8, ...

Ratio:

×1.2

This is geometric.

This distinction is extremely important when choosing a model.

Patterns in Social Media

Suppose one person shares a message with 3 people, and each of those people shares it with 3 more.

The number of new people reached at each stage could be modelled as:

1, 3, 9, 27, 81, ...

The rule is:

multiply by 3

This illustrates why information can sometimes spread extremely rapidly through networks.

Real social networks are much more complicated because people may share with the same individuals, stop sharing, or have different numbers of contacts.

Again, the sequence is a simplified model.

Patterns in Construction

Suppose a staircase design uses:

• 4 blocks in the first level
• 7 blocks in the second
• 10 blocks in the third
• 13 blocks in the fourth

The sequence is:

4, 7, 10, 13, ...

The common difference is:

3

The explicit rule is:

aₙ = 3n + 1

For level 30:

a₃₀ = 3(30) + 1

a₃₀ = 91

A builder could use the sequence to predict the number of blocks needed for a particular level.

Patterns in Technology

Sequences are important in computing and technology.

They can describe:

• repeated calculations
• computer algorithms
• memory usage
• data growth
• network traffic
• digital image sizes
• storage requirements

For example, digital storage sizes often follow repeated powers of 2:

1, 2, 4, 8, 16, 32, 64, ...

The sequence doubles each time.

Predicting Future Outcomes

One of the main reasons for creating a sequence model is to make predictions.

Suppose electricity consumption follows:

120, 135, 150, 165, 180, ...

If the pattern continues, we can predict:

Term 6 = 195

Term 7 = 210

Term 8 = 225

But every prediction contains an assumption:

the existing pattern will continue.

This assumption should always be considered when interpreting the result.

Interpolation and Extrapolation

Suppose we have data for years 2022–2026.

Using the pattern to estimate a missing value for 2024 is an example of interpolation because we are estimating within the range of known data.

Using the pattern to predict 2035 is extrapolation because we are extending beyond the known data.

Extrapolation is generally more uncertain because the pattern may change.

Worked Example: Savings

Maya has $300 saved and adds $75 each month.

The sequence is:

300, 375, 450, 525, 600, ...

Common difference:

d = 75

Explicit rule:

aₙ = 300 + 75(n − 1)

Find the amount at the 12th stage:

a₁₂ = 300 + 75(11)

a₁₂ = 300 + 825

a₁₂ = $1125

If Maya continues following the plan, the model predicts $1,125.

Worked Example: Population

A population begins with 400 organisms and increases by 25% during each time period.

Multiplier:

1.25

Sequence:

400, 500, 625, 781.25, ...

Explicit model:

aₙ = 400(1.25)ⁿ⁻¹

At the sixth stage:

a₆ = 400(1.25)⁵

a₆ ≈ 1220.7

Depending on the context, we might predict approximately 1,221 organisms.

Worked Example: Declining Value

A machine is initially worth $10,000 and loses 10% of its value each year.

It retains:

90% = 0.90

of its value.

The sequence begins:

10000, 9000, 8100, 7290, ...

Explicit model:

aₙ = 10000(0.90)ⁿ⁻¹

This allows future values to be predicted.

Choosing a Suitable Model

When examining a real-world pattern, ask:

What quantity is changing?

Then determine:

Does it change by approximately the same amount?

If yes, consider an arithmetic sequence.

If not, ask:

Does it change by approximately the same factor or percentage?

If yes, consider a geometric sequence.

Then ask:

Is there enough evidence to assume the pattern will continue?

Mathematics can produce an answer even when the model itself is unrealistic. Good modelling requires us to evaluate both.

Model vs. Reality

Suppose a bacteria population doubles every hour.

Starting with 100 bacteria:

100, 200, 400, 800, 1600, ...

After many hours, the mathematical model predicts enormous populations.

But eventually:

• food becomes limited
• waste accumulates
• space becomes limited
• environmental conditions change

The original model stops being realistic.

This illustrates an important idea:

Mathematical predictions are only as reliable as the assumptions behind the model.

Real Data May Not Be Perfect

Real-world sequences do not always follow exact rules.

Suppose a city's population is:

52,000, 53,100, 54,350, 55,400, 56,700

The increases are not identical.

Real data often contains variation.

A mathematical model may therefore describe an overall trend rather than reproduce every value exactly.

This is normal in science, economics, engineering, and social research.

Creating a Sequence Model

Suppose a fitness program begins with a 2 km run and increases the distance by 0.5 km each week.

The sequence is:

2, 2.5, 3, 3.5, 4, ...

Recursive rule:

a₁ = 2

aₙ = aₙ₋₁ + 0.5

Explicit rule:

aₙ = 2 + 0.5(n − 1)

After 10 stages:

a₁₀ = 2 + 0.5(9)

a₁₀ = 6.5 km

The sequence turns a real situation into a mathematical model that can be used for prediction.

Common Mistakes

Assuming every real-world pattern is exact

Real data often contains variation.

Confusing constant increase with percentage increase

Adding $100 every year is arithmetic.

Increasing by 10% every year is geometric.

Assuming a pattern will continue forever

Environmental, economic, technological, or social conditions can change.

Ignoring the meaning of the terms

Always identify what n represents: days, years, rows, generations, payments, or something else.

Using the wrong starting point

If a₁ represents the initial value, then the exponent in a geometric sequence is usually n − 1.

Treating a model as reality

A mathematical model is a simplified representation. Its predictions depend on its assumptions.

Key Terms

Sequence — an ordered list of values following a pattern or rule.

Mathematical model — a mathematical representation of a real-world situation.

Arithmetic sequence — a sequence with a constant difference between consecutive terms.

Common difference — the amount repeatedly added or subtracted in an arithmetic sequence.

Geometric sequence — a sequence with a constant ratio between consecutive terms.

Common ratio — the factor by which consecutive terms of a geometric sequence are multiplied.

Recursive rule — a rule that generates a term from one or more previous terms.

Explicit rule — a formula that calculates a term directly from its position.

Prediction — an estimate of a future or unknown value based on a model.

Interpolation — estimating a value within the range of known data.

Extrapolation — predicting a value beyond the range of known data.

Fibonacci sequence — a sequence in which each new term is the sum of the previous two terms.

Growth model — a mathematical representation of a quantity increasing over time.

Depreciation — a decrease in the value of an asset over time.

Assumption — a condition accepted when constructing or using a mathematical model.

Key Takeaways

• Sequences occur throughout nature, society, finance, science, and technology.
• Arithmetic sequences model repeated constant changes.
• Geometric sequences model repeated multiplication or percentage changes.
• Population growth can sometimes be represented using sequences.
• Savings, investments, and depreciation provide common financial applications.
• Fibonacci numbers appear in some patterns of biological growth and arrangement.
• Recursive rules show how a real-world quantity changes from one stage to the next.
• Explicit rules allow distant values to be predicted directly.
• Sequences can help predict future outcomes when the underlying pattern continues.
• Real-world patterns are often approximate rather than exact.
• Extrapolating far beyond known data can produce unreliable predictions.
• A mathematical model should always be evaluated in terms of its assumptions and limitations.

Check Your Understanding

1. Give three examples of situations in nature or society that could be represented using sequences.
2. A student saves $40 every week. Write the first six terms representing the total amount saved.
3. Write a recursive rule and an explicit rule for the sequence in Question 2.
4. A bacteria population begins at 300 and doubles during each time interval. Write the first six terms.
5. Write an explicit rule for the bacteria population in Question 4.
6. A phone is initially worth $1,200 and retains 75% of its value each year. Write the first four terms of the value sequence.
7. Is the phone-value sequence arithmetic or geometric? Explain.
8. A theatre has 24 seats in its first row and 4 additional seats in each following row. How many seats are in the 30th row?
9. An investment begins at $5,000 and grows by 6% each year. Write a sequence model for its value.
10. Explain why a population-growth sequence might make reasonable predictions for the next few years but unrealistic predictions for the next 500 years.
11. Explain the difference between interpolation and extrapolation.
12. Find a numerical pattern from nature, finance, technology, or everyday life. Describe the sequence, identify its rule, and explain what the pattern could help you predict.