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

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.

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.

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.

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.