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Number sequences: find the rule fast
A checking order that cracks most sequence questions in under a minute: differences, ratios, mixed operations, interleaving and the families worth knowing.
The checking order
- 1
Differences
Subtract each term from the next. If those aren't constant, take differences again (second differences).
- 2
Ratios
Divide each term by the one before. Constant ratios mean a geometric sequence.
- 3
Mixed operations
Try "× a, then + b" when terms grow a bit faster than doubling or tripling.
- 4
Interleaving
If terms jump up and down, split into odd and even positions: two simpler sequences.
- 5
Known families
Squares, cubes, primes, and "add the previous two" (Fibonacci-style).
Worked examples
Worked example · Second differences
Checked by code3, 7, 13, 21, 31, ?
Show the working
- 1.Differences: 4, 6, 8, 10. They rise by 2 each time.
- 2.Next difference 12: 31 + 12 = 43.
Answer43.
Worked example · Mixed operations
Checked by code3, 7, 15, 31, ?
Show the working
- 1.Each term is double the last plus 1: 3 × 2 + 1 = 7, 7 × 2 + 1 = 15, 15 × 2 + 1 = 31.
- 2.31 × 2 + 1 = 63.
Answer63.
Worked example · Interleaved
Checked by code1, 10, 3, 20, 5, 30, ?
Show the working
- 1.Odd positions: 1, 3, 5. Even positions: 10, 20, 30.
- 2.The next term is in an odd position: 7.
Answer7.
Worked example · Squares plus one
Checked by code2, 5, 10, 17, 26, ?
Show the working
- 1.Differences 3, 5, 7, 9 are the odd numbers, a sign of squares.
- 2.The terms are n² + 1: the next is 6² + 1 = 37.
Answer37.
Common mistakes
Common mistakes
- Stopping at first differences when the second differences are constant.
- Missing interleaving when terms alternate up and down.
- Spending three minutes on one question. If nothing fits in about a minute, guess sensibly and move on.
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