12.2 Term-to-Term Rules

12.2 Term-to-Term Rules

A term-to-term rule tells us how to get from one term in a sequence to the next.

For example:

2, 5, 8, 11, 14, …

Each term is found by adding 3 to the previous term.

So the term-to-term rule is:

Add 3

Term-to-term rules may involve:

  • addition
  • subtraction
  • multiplication
  • division
  • alternating operations
  • more complicated changing patterns

Recognising the correct rule is one of the first skills you need when solving sequence questions.

Concept

A sequence is an ordered list of numbers that follows a pattern.

For example:

2, 5, 8, 11, 14, …

To understand how this sequence is formed, compare consecutive terms.

2 → 5

5 → 8

8 → 11

11 → 14

Each time:

+3

Therefore:

Term-to-term rule = add 3

This means that if you know one term, you can find the next term by adding 3.

What Does Term-to-Term Mean?

The words term-to-term simply mean:

from one term to the next term

For example:

7, 12, 17, 22, 27, …

Start with 7.

7 + 5 = 12

12 + 5 = 17

17 + 5 = 22

22 + 5 = 27

Therefore:

Term-to-term rule = add 5

1. Adding the Same Number

Consider:

2, 5, 8, 11, 14, …

Find the difference between consecutive terms:

5 – 2 = 3

8 – 5 = 3

11 – 8 = 3

14 – 11 = 3

The same number is being added each time.

Therefore:

Term-to-term rule = add 3

The next term is:

14 + 3 = 17

So the sequence continues:

2, 5, 8, 11, 14, 17, …

Example Math

Find the next two terms:

6, 10, 14, 18, …

Check the difference:

10 – 6 = 4

14 – 10 = 4

18 – 14 = 4

Therefore:

Term-to-term rule = add 4

Continue:

18 + 4 = 22

22 + 4 = 26

Answer:

22, 26

2. Subtracting the Same Number

Consider:

20, 16, 12, 8, 4, …

Each term becomes smaller.

Check the difference:

16 – 20 = -4

12 – 16 = -4

8 – 12 = -4

4 – 8 = -4

Therefore:

Term-to-term rule = subtract 4

The next term is:

4 – 4 = 0

So the sequence continues:

20, 16, 12, 8, 4, 0, …

Example Math

Find the next two terms:

35, 29, 23, 17, …

Difference:

29 – 35 = -6

23 – 29 = -6

17 – 23 = -6

So:

Term-to-term rule = subtract 6

Continue:

17 – 6 = 11

11 – 6 = 5

Answer:

11, 5

3. Multiplying by the Same Number

Not every sequence uses addition or subtraction.

Consider:

3, 6, 12, 24, 48, …

Check what happens:

3 × 2 = 6

6 × 2 = 12

12 × 2 = 24

24 × 2 = 48

Therefore:

Term-to-term rule = multiply by 2

The next term is:

48 × 2 = 96

Example Math

Find the next two terms:

5, 15, 45, 135, …

Check:

5 × 3 = 15

15 × 3 = 45

45 × 3 = 135

Therefore:

Term-to-term rule = multiply by 3

Continue:

135 × 3 = 405

405 × 3 = 1215

Answer:

405, 1215

4. Dividing by the Same Number

Consider:

81, 27, 9, 3, 1, …

Check:

81 ÷ 3 = 27

27 ÷ 3 = 9

9 ÷ 3 = 3

3 ÷ 3 = 1

Therefore:

Term-to-term rule = divide by 3

Example Math

Find the next two terms:

160, 80, 40, 20, …

Check:

160 ÷ 2 = 80

80 ÷ 2 = 40

40 ÷ 2 = 20

Therefore:

Term-to-term rule = divide by 2

Continue:

20 ÷ 2 = 10

10 ÷ 2 = 5

Answer:

10, 5

How to Find a Term-to-Term Rule

When you are given a sequence, follow this order.

Step 1: Check the Differences

Subtract consecutive terms.

For example:

4, 9, 14, 19, …

Differences:

+5, +5, +5

So:

Add 5

Step 2: Check Multiplication or Division

If the differences are not constant, check whether each term is multiplied or divided by the same number.

Example:

4, 12, 36, 108, …

4 × 3 = 12

12 × 3 = 36

36 × 3 = 108

So:

Multiply by 3

Step 3: Look for a Different Pattern

If neither of these works, the sequence may involve:

  • alternating operations
  • changing differences
  • square numbers
  • cube numbers
  • previous-term relationships
  • quadratic sequences

Do not assume every sequence has one simple operation.

Common Difference

If a sequence increases or decreases by the same amount each time, that amount is called the common difference.

For example:

2, 5, 8, 11, 14, …

The differences are:

+3, +3, +3, +3

Therefore:

Common difference = 3

We often use:

d

to represent the common difference.

So:

d = 3

Negative Common Difference

Consider:

25, 20, 15, 10, 5, …

Difference:

20 – 25 = -5

Therefore:

d = -5

This tells us the sequence decreases by 5 each time.

Term-to-Term Rule vs Common Difference

These ideas are closely related, but they are not exactly the same wording.

For:

7, 11, 15, 19, …

the term-to-term rule is:

Add 4

and:

d = 4

For:

20, 17, 14, 11, …

the term-to-term rule is:

Subtract 3

and:

d = -3

Term-to-Term Rule vs nth Term

This is very important.

A term-to-term rule tells you how to get from one term to the next.

An nth-term rule allows you to calculate a term directly from its position.

Consider:

2, 5, 8, 11, 14, …

Term-to-term rule:

Add 3

nth term:

3n – 1

These are not the same thing.

Why Is the nth Term Useful?

Suppose you want the 100th term of:

2, 5, 8, 11, 14, …

Using the term-to-term rule would mean adding 3 again and again.

That would take too long.

Instead, use:

3n – 1

For:

n = 100

3(100) – 1

= 300 – 1

= 299

So the 100th term is:

299

Alternating Term-to-Term Rules

Some sequences use two rules repeatedly.

For example:

2, 5, 4, 7, 6, 9, …

Look carefully:

2 → 5 = +3

5 → 4 = -1

4 → 7 = +3

7 → 6 = -1

6 → 9 = +3

So the rule alternates:

+3, -1, +3, -1, …

The next term is:

9 – 1 = 8

Example Math

Find the next two terms:

10, 20, 18, 36, 34, …

Look at the operations:

10 × 2 = 20

20 – 2 = 18

18 × 2 = 36

36 – 2 = 34

So the alternating rule is:

×2, -2

Continue:

34 × 2 = 68

68 – 2 = 66

Answer:

68, 66

Changing Differences

Some sequences have differences that change.

Consider:

1, 4, 9, 16, 25, …

First differences:

3, 5, 7, 9

The differences are not constant.

However, they increase by 2 each time.

This is the familiar square-number sequence:

1², 2², 3², 4², 5², …

So you should not describe its term-to-term rule simply as “add the same number.”

Fibonacci-Type Rules

Sometimes the next term depends on previous terms.

For example:

1, 1, 2, 3, 5, 8, 13, …

Here:

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

The term-to-term rule is:

Add the previous two terms

This is different from simply adding the same constant each time.

Example Math

Example 1

Find the next term:

9, 14, 19, 24, …

Difference:

+5

Therefore:

24 + 5 = 29

Answer:

29

Example 2

Find the next term:

50, 43, 36, 29, …

Difference:

-7

Therefore:

29 – 7 = 22

Answer:

22

Example 3

Find the next term:

4, 12, 36, 108, …

Rule:

×3

Therefore:

108 × 3 = 324

Answer:

324

Example 4

Find the next term:

625, 125, 25, 5, …

Rule:

÷5

Therefore:

5 ÷ 5 = 1

Answer:

1

Example 5

Find the missing term:

11, 17, 23, __, 35

Difference:

+6

Therefore:

23 + 6 = 29

Answer:

29

Exam Tip

When you see a sequence, do not guess the pattern from only the first two terms.

Check at least three or four transitions.

For example:

2, 4, 8, 16, …

You might first notice:

2 → 4 = +2

But:

4 → 8 = +4

So “add 2” cannot be the rule.

Instead:

2 × 2 = 4

4 × 2 = 8

8 × 2 = 16

Therefore:

Term-to-term rule = multiply by 2

A/A* Exam Technique

For difficult sequence questions, write the operation between each pair of terms.

For example:

3 → 9 → 27 → 81

Write:

×3 ×3 ×3

This makes the pattern much easier to see.

For:

18 → 13 → 8 → 3

write:

-5 -5 -5

And for an alternating pattern:

4 → 8 → 7 → 14 → 13

write:

×2 -1 ×2 -1

This is faster and safer than trying to recognise the whole sequence mentally.

Common Mistakes

Mistake 1: Looking Only at the First Pair of Terms

Consider:

3, 6, 12, 24, …

You might say:

3 → 6 means add 3.

But:

6 + 3 ≠ 12

The actual rule is:

Multiply by 2

Always check the rule against several terms.

Mistake 2: Confusing +3 with ×3

Consider:

2, 6, 18, 54, …

This is not:

+3

because:

6 + 3 = 9

Instead:

6 × 3 = 18

So the rule is:

Multiply by 3

Mistake 3: Ignoring Negative Numbers

Consider:

7, 3, -1, -5, …

The rule continues through zero.

Difference:

3 – 7 = -4

-1 – 3 = -4

So:

Subtract 4

Mistake 4: Calling Every Difference a Common Difference

A common difference exists only when the difference is constant.

For:

1, 4, 9, 16, …

differences are:

3, 5, 7

So there is no constant common difference.

Mistake 5: Confusing Term-to-Term Rule and nth Term

For:

2, 5, 8, 11, …

Term-to-term rule:

Add 3

nth term:

3n – 1

Do not write “3n – 1” when the question specifically asks for the term-to-term rule.

Question & Answer

1. What is a term-to-term rule?

It tells you how to get from one term of a sequence to the next.

2. What is the term-to-term rule for:

3, 8, 13, 18, …

Add 5.

3. What is the term-to-term rule for:

30, 24, 18, 12, …

Subtract 6.

4. What is the term-to-term rule for:

2, 10, 50, 250, …

Multiply by 5.

5. What is the term-to-term rule for:

64, 32, 16, 8, …

Divide by 2.

6. What does common difference mean?

It is the fixed difference between consecutive terms of an arithmetic sequence.

7. Is a term-to-term rule the same as an nth-term rule?

No.

The term-to-term rule tells you how to move from one term to the next.

The nth term tells you how to calculate a term directly using its position.

8. Can a sequence have more than one operation?

Yes.

Some sequences use alternating operations such as:

+4, -2, +4, -2, …

9. Must every sequence have a constant difference?

No.

Quadratic and other non-linear sequences do not have a constant first difference.

10. What should I check first?

First check whether the same number is being:

added or subtracted.

If not, check whether the terms are being:

multiplied or divided.

Practice Questions

Try these without looking at the answers.

1

Find the next two terms:

5, 9, 13, 17, …

2

Find the next two terms:

40, 33, 26, 19, …

3

Find the next term:

2, 8, 32, 128, …

4

Find the next term:

243, 81, 27, 9, …

5

State the term-to-term rule:

7, 12, 17, 22, …

6

State the term-to-term rule:

80, 72, 64, 56, …

7

State the term-to-term rule:

6, 18, 54, 162, …

8

Find the missing term:

15, 20, 25, __, 35

9

Find the next two terms:

3, 6, 5, 10, 9, …

10

Find the next term:

1, 4, 9, 16, 25, …

Answers

1

21, 25

Rule:

Add 4

2

12, 5

Rule:

Subtract 7

3

512

Rule:

Multiply by 4

4

3

Rule:

Divide by 3

5

Add 5

6

Subtract 8

7

Multiply by 3

8

30

9

18, 17

Alternating rule:

×2, -1

10

36

This is the square-number sequence.

Want to Know More?

After understanding term-to-term rules, continue with:

  • Finding the nth Term
  • Arithmetic Sequences
  • Linear Sequences
  • Common Difference
  • Geometric Sequences
  • Quadratic Sequences
  • Difference Tables
  • Special Number Sequences
  • Finding Missing Terms
  • Determining Whether a Number Is in a Sequence

These topics build directly on your ability to recognise how one term changes into the next.

Final Exam Reminder

When you see a sequence, ask:

How do I get from one term to the next?

Check in this order:

Addition → Subtraction → Multiplication → Division → More complex pattern

For example:

2, 5, 8, 11, 14, …

Add 3

20, 16, 12, 8, 4, …

Subtract 4

3, 6, 12, 24, 48, …

Multiply by 2

81, 27, 9, 3, 1, …

Divide by 3

And remember the key distinction:

Term-to-term rule = how one term changes into the next.

nth term = how to find a term directly from its position, n.