Number sequences

The idea

A sequence is a list of numbers following a rule. Your first job is always to find the gap between neighbouring terms: subtract each term from the next one. If every gap is the same, the rule is 'add (or subtract) that amount each time' — this is a linear sequence.

If the gaps themselves grow in a pattern, look again: gaps that double suggest the terms are doubling, and sequences like square numbers (1, 4, 9, 16…) or triangular numbers have gaps that increase by 1 each time.

To find a term far ahead in a linear sequence, count how many jumps you need from a term you know, multiply by the gap, and add on. To fill a missing middle term, check the rule works on both sides of the gap.

Worked example

Here is a sequence: 4, 9, 14, 19, 24, … What are the next two terms?

1. Find the gaps: 9 − 4 = 5, 14 − 9 = 5, 19 − 14 = 5, 24 − 19 = 5.
2. Every gap is 5, so the rule is 'add 5 each time'.
3. Next term: 24 + 5 = 29.
4. Term after that: 29 + 5 = 34.

Answer: 29 and 34

Watch out

• Assuming the first gap is the rule without checking it stays the same for every pair of terms.
• For sequences that go down, adding the gap instead of subtracting it.
• Missing that gaps can grow — in 3, 6, 12, 24 the terms double, so adding a fixed amount gives the wrong answer.