The checklist
Work through these in order and stop at the first that fits every term.
Differences. Subtract each term from the next. If the differences are constant or form their own simple pattern (2, 4, 6, 8…), you have the rule.
Ratios. Divide each term by the one before. Constant ratios mean multiplication.
Mixed operations. ×2 + 1, ×3 − 2 and similar rules show up as ratios that are "almost" constant.
Known sequences. Squares, cubes, primes, Fibonacci (each term is the sum of the previous two), n² ± 1.
Alternating series. Two interleaved series: look at the 1st, 3rd, 5th… terms separately from the 2nd, 4th, 6th….
Worked examples
Series | Rule | Next |
|---|---|---|
2, 6, 12, 20, 30 | differences 4, 6, 8, 10 | 42 |
5, 11, 23, 47 | ×2 + 1 | 95 |
120, 99, 80, 63, 48 | 11² − 1, 10² − 1, 9² − 1 … | 35 |
1, 1, 2, 3, 5, 8 | Fibonacci | 13 |
2, 3, 5, 7, 11, 13 | primes | 17 |
Letter series
Convert letters to positions: A = 1, B = 2, … Z = 26. The reverse positions (A = 26 … Z = 1) are also useful. A handy memory aid is EJOTY = 5, 10, 15, 20, 25.
Series | Positions | Next |
|---|---|---|
A, C, F, J, O | 1, 3, 6, 10, 15 (gaps 2, 3, 4, 5) | U (21) |
AZ, BY, CX | first letter +1, second letter −1 | DW |
Finding the wrong term
Work out the rule from the majority of terms, then find the one that breaks it.
Example. 4, 8, 16, 32, 62, 128 doubles each time, so 62 should be 64.
Shortcuts and tricks
The standard way is to write the differences between terms, then the differences of those. These patterns are worth recognising at sight.
Products of neighbours
Example: 6, 12, 20, 30, 42, ?
Standard way: the differences are 6, 8, 10, 12, so the next is 42 + 14 = 56.
Shortcut: the terms are 2 × 3, 3 × 4, 4 × 5, 5 × 6, 6 × 7, so the next is 7 × 8 = 56.
Add or subtract 1, then look again
Example: 3, 7, 15, 31, ?
Standard way: each term is double the previous plus 1: 31 × 2 + 1 = 63.
Shortcut: add 1 to each term: 4, 8, 16, 32 are powers of 2, so the next is 64 − 1 = 63.
Squares and cubes nearby
Numbers like 26, 63 and 124 are cubes plus or minus 1, and 35, 48 and 99 are squares minus 1. Check these before working out differences.
Letter series by position
Example: B, E, I, N, ?
Standard way: count through the alphabet: B to E is 3, E to I is 4, I to N is 5, so the next jump is 6: T.
Shortcut: use EJOTY (E = 5, J = 10, O = 15, T = 20, Y = 25). N = 14, and 14 + 6 = 20 is T.
Exam tips
Test the rule on every term, not just the first two.
Large jumps suggest multiplication or squares; small, growing gaps suggest addition.
If nothing works in 30 seconds, try an alternating pattern.