Learn · Numeracy · 5 min

Patterns: find the rule

A pattern question gives you a few terms and asks for one you can’t see. The trick is never to keep counting — find the step, then jump:

stephow much it changes each time — subtract two neighbours
ruleterm n = step × n + a fix-up number
jumpput any n into the rule — term 100 takes one line

See it: every step-by-step pattern has one rule

4+37+310+313+316+319…

Starts at: 4

Goes up by: 3

Rule: term n = 3 × n + 1

Jump to term 10 — no counting

term 10 = 3 × 10 + 1 = 31

Now — the one move

Pattern: 7, 11, 15, 19, … What is the 50th term?

Step × n, then fix the offset

① Step = 11 − 7 = 4, so the rule starts 4 × n
② Test on term 1: 4 × 1 = 4, but the first term is 7 — that’s 3 too small
③ So the rule is 4n + 3. Check term 2: 4 × 2 + 3 = 11 ✓
④ Term 50 = 4 × 50 + 3 = 203

Tips & tricks

Going DOWN? The step is negative

20, 17, 14, … → step = −3, rule = −3n + 23. Term 5 = −15 + 23 = 8.

Not adding the same each time? Try multiplying

3, 6, 12, 24 doubles — that’s × 2 each time, not + 3. Always check both.

Growing patterns of shapes — count the extra bits

If each new shape adds 3 matchsticks and the first uses 4, the rule is 3n + 1.

“Which term equals 87?” — run the rule backwards

With 4n + 3 = 87 → 4n = 84 → n = 21. If n isn’t a whole number, 87 is not in the pattern.

Always test your rule on a term you can see

Ten seconds of checking catches an off-by-one before it costs you the mark.

Now you try — 0/4

1. Pattern: 2, 6, 10, 14, … What is the 7th term?

2. Which rule gives the pattern 5, 8, 11, 14, … ?

3. Pattern: 2, 5, 8, 11, … What is the 101st term?

4. In the pattern 4, 9, 14, 19, … which term is equal to 59?

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