Number Patterns & Rules
Discover how to describe, extend, and find the rules behind number patterns. Patterns are everywhere in maths!
What Are Number Patterns?
A number pattern (or sequence) is a list of numbers that follow a particular rule. Once you find the rule, you can work out any number in the pattern.
Growing Patterns
Numbers get bigger each time.
3, 7, 11, 15, 19, ...
Rule: add 4 each time
Shrinking Patterns
Numbers get smaller each time.
100, 90, 80, 70, 60, ...
Rule: subtract 10 each time
Multiplying Patterns
Numbers are multiplied by the same amount each step.
2, 6, 18, 54, 162, ...
Rule: multiply by 3 each time
Finding the Rule
To find the rule, look at the difference between each pair of numbers. If the difference is always the same, the pattern uses addition or subtraction.
Step-by-step: Find the rule for 5, 12, 19, 26, 33
Rule: Start at 5, add 7 each time. The next number would be 33 + 7 = 40.
Using a Table to Find Rules
An input-output table shows the relationship between a position number and its value. You can write a rule using algebra.
| Position (n) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Value | 4 | 7 | 10 | 13 | 16 |
Step 1: The difference between values is always 3 (add 3 each time).
Step 2: Try the rule: value = 3 × n + something.
Step 3: When n = 1, value = 4. So 3 × 1 + ? = 4, meaning ? = 1.
Rule: value = 3n + 1
Two-Step Rules
Some patterns use a two-step rule where you multiply (or divide) AND add (or subtract). The general form is:
value = a × n + b
where a is the common difference and b is a constant.
Example: 5, 9, 13, 17, 21
Common difference = 4, so a = 4.
When n = 1: 4 × 1 + b = 5, so b = 1.
Rule: 4n + 1
Check: n = 3 → 4 × 3 + 1 = 13. Correct!
Key Vocabulary
Sequence
An ordered list of numbers that follows a particular rule.
Term
Each individual number in a sequence (e.g. the 3rd term).
Common Difference
The constant amount added or subtracted between terms.
Rule (Formula)
An algebraic expression that lets you find any term (e.g. 3n + 2).
Worked Examples
Find the next two numbers: 8, 15, 22, 29, ...
Step 1: Find the difference: 15 − 8 = 7, 22 − 15 = 7, 29 − 22 = 7.
Step 2: Rule is add 7 each time.
Answer: 29 + 7 = 36, 36 + 7 = 43
Write the rule for: Position 1 = 6, Position 2 = 11, Position 3 = 16, Position 4 = 21
Step 1: Common difference = 11 − 6 = 5, so a = 5.
Step 2: When n = 1: 5 × 1 + b = 6, so b = 1.
Rule: 5n + 1
Using the rule 2n + 3, find the 10th term.
Step 1: Substitute n = 10 into 2n + 3.
Step 2: 2 × 10 + 3 = 20 + 3
Answer: The 10th term is 23
Knowledge Check
Select the correct answer for each question.
Question 1
What is the next number in this pattern? 4, 9, 14, 19, ...
Question 2
What is the rule for this pattern? 3, 7, 11, 15, 19, ...
Question 3
Using the rule 3n + 2, what is the 5th term?
Question 4
A pattern starts at 2 and each term is multiplied by 3. What is the 4th term?
Question 5
The rule for a pattern is 4n − 1. Which sequence does it produce?
Key Concepts Summary
- ●A number pattern is a sequence of numbers following a rule.
- ●Find the common difference by subtracting consecutive terms.
- ●Use the form an + b to write a rule for any position.
- ●Substitute the position number into the rule to find any term.
- ●Always check your rule works for more than one term.