What is a number pattern?
A number pattern (or sequence) is a list of numbers that follows a rule. Once you know the rule, you can say what comes next, or even what the 100th number will be, without writing out every number in between.
Look at 2, 5, 8, 11, 14, … Each number is 3 more than the one before it. So the next numbers are 17, 20 and 23.
Mathematics is often described as the study of patterns. Spotting them is a skill you will use in algebra, in geometry and in everyday problem solving.
How to find the rule
Ask these questions, in this order:
Is the same number added or subtracted each time? Look at the differences between neighbours. 7, 12, 17, 22 has a difference of 5 every time.
Is each number multiplied or divided by the same amount? 3, 6, 12, 24, 48 doubles each time.
Do the differences themselves form a pattern? In 1, 2, 4, 7, 11, 16, the differences are 1, 2, 3, 4, 5.
Is it a special sequence you already know? Squares, cubes and triangular numbers appear again and again.
Example. Find the next two numbers: 1, 3, 9, 27, …
Each number is 3 times the one before it. So the next numbers are 27 × 3 = 81 and 81 × 3 = 243.
Famous sequences
Name | Sequence | Rule |
|---|---|---|
Counting numbers | 1, 2, 3, 4, 5, … | add 1 |
Odd numbers | 1, 3, 5, 7, 9, … | add 2, starting at 1 |
Even numbers | 2, 4, 6, 8, 10, … | add 2, starting at 2 |
Square numbers | 1, 4, 9, 16, 25, … | 1×1, 2×2, 3×3, … |
Triangular numbers | 1, 3, 6, 10, 15, … | add 2, then 3, then 4, … |
Cube numbers | 1, 8, 27, 64, 125, … | 1×1×1, 2×2×2, … |
Powers of 2 | 1, 2, 4, 8, 16, 32, … | double each time |
Patterns you can see
Many sequences get their names from the shapes they make with dots.
Square numbers. Arrange dots in a square: 1 dot, then a 2 by 2 square (4 dots), then 3 by 3 (9 dots). The number of dots is a square number.
Triangular numbers. Arrange dots in rows of 1, 2, 3, …: 1, then 1 + 2 = 3, then 1 + 2 + 3 = 6, then 10. These are triangular numbers.
Patterns hidden in sums
Add the odd numbers one at a time:
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
The answers are the square numbers. The sum of the first n odd numbers is n × n. So the sum of the first 10 odd numbers is 10 × 10 = 100, and you never have to add them one by one.
Two neighbouring triangular numbers also add up to a square: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16.
Finding a term far along
When the same number is added each time, you can jump straight to any position.
Example. The pattern 4, 7, 10, 13, … adds 3 each time. What is the 20th number?
To get from the 1st number to the 20th, you add 3 a total of 19 times. So the 20th number is 4 + 19 × 3 = 4 + 57 = 61.
Common mistakes
Checking only the first two numbers. 2, 4, 8 could be "add 2" or "double". Always check the rule on every pair.
Counting the jumps wrongly. From the 1st to the 20th term there are 19 jumps, not 20.
Forgetting that a rule can be "add a growing amount". Look at the differences when nothing obvious works.
Summary
A number pattern follows a rule; find it by looking at differences, ratios or known sequences.
Square numbers are 1, 4, 9, 16, …; triangular numbers are 1, 3, 6, 10, ….
The sum of the first n odd numbers is n × n.
In an "add the same amount" pattern, the nth term is the first term plus (n − 1) jumps.