Patterns and rules · Usually met around P4
Rules for number sequences
Look at the gaps; if they change, find their pattern.
How to do it
Method
Look at the gaps; if they change, find their pattern
1, 3, 6, 10: gaps 2, 3, 4 → next 10 + 5 = 15
Another way
Try multiplying
3, 6, 12, 24: × 2 each time → 48
In full
First see how far apart neighbouring numbers are. If it is the same amount added (or taken away) every time, the rule is “add 4 each time” or “subtract 6 each time”. If the gaps keep growing, try multiplying: in 3, 6, 12, 24 each number is double the one before. If the gaps themselves follow a pattern, look at the gaps: 1, 3, 6, 10 has gaps 2, 3, 4, so the next gap is 5 and the next number 15; these are the triangular numbers. 1, 4, 9, 16 are 1 × 1, 2 × 2, 3 × 3, 4 × 4, the square numbers, with gaps 3, 5, 7. Once you have a rule, check it on every pair, not just the first two.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Fill in the sequence: 15, □, 31, □, 47
Fill in each box
Show the answer and working
Answer: 23，39
From 15 to 31 is two steps adding 16, so each step adds 8: 15, 23, 31, 39, 47.
Example 2
Continue the sequence: 1, 4, 9, 16, □
Show the answer and working
Answer: 25
The gaps are 3, 5, 7, each 2 more, so the next gap is 9: 16 + 9 = 25. They are square numbers: 25 = 5 × 5.
Example 3
What is the rule for 5, 15, 45, 135?
- A. Multiply by 3 each time
- B. Add 10 each time
- C. Add 30 each time
- D. Multiply by 4 each time
Choose the right answer
Show the answer and working
Answer: A. Multiply by 3 each time
5 × 3 = 15, 15 × 3 = 45, 45 × 3 = 135: every pair is × 3.
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