60-second revision
- Always start with the differences between terms; if they don't fit, take the differences of the differences.
- Slow growth means addition; fast growth means multiplication; very fast growth means a growing multiplier (×2, ×3, ×4...).
- Terms close to squares or cubes (63, 80, 99 or 7, 26, 63) are usually n² ± k or n³ ± k.
- A long series (7–8 terms) that jumps up and down is usually two series interleaved: odd and even positions.
- Mixed rule ×a + b: check whether each term ~ previous × a, then find the constant left over.
- Primes to 100: 25 of them. 2 is the only even prime; 1 is not prime; 51, 57, 87 and 91 are not prime.
- Wrong-number-out: find the rule from most terms, then test each term; exactly one fails.
The attack plan (use this order)
Most exam series are solved within 30 seconds if you test patterns in a fixed order instead of guessing. Write the differences under the series first; that one step solves about half of all questions.
- First differences: subtract each term from the next. Constant? It is an arithmetic series.
- Second differences: if the first differences change, take their differences. A constant or a clear pattern here solves it.
- Ratios: if the series grows fast, divide each term by the previous one. Look for a constant ratio or a growing multiplier.
- Squares and cubes: compare each term with the nearest square or cube. Look for n², n³, n² ± k, n³ ± k, or differences that are squares or cubes.
- Primes: check whether the terms (or the differences) are consecutive primes.
- Alternating or interleaved: if the series zig-zags or is long, split it into odd positions and even positions.
- Mixed operations: test ×a + b, ×a − b, or alternating operations (×2, +3, ×2, +3).
- Wrong-number-out: find the rule that fits the majority, then locate the one term that breaks it.
Pattern table with worked examples
| Pattern | How to spot it | Example | Answer |
|---|---|---|---|
| Constant difference | Same gap every time | 4, 11, 18, 25, ? | 32 (+7 each time) |
| Differences rise steadily | Gaps 2, 4, 6, 8... | 3, 5, 9, 15, 23, ? | 33 (next gap 10) |
| Second differences | Gaps change, gaps of gaps are steady | 4, 6, 12, 24, 44, ? | 74 (gaps 2, 6, 12, 20, 30; second gaps 4, 6, 8, 10) |
| Differences are cubes | Gaps 1, 8, 27, 64... | 10, 11, 19, 46, 110, ? | 235 (gap 125) |
| Constant ratio | Each term = previous × same number | 16, 24, 36, 54, ? | 81 (×1.5) |
| Growing multiplier | ×1, ×2, ×3, ×4... | 2, 2, 4, 12, 48, ? | 240 (×5) |
| Shrinking divisor | ÷6, ÷5, ÷4... | 720, 120, 24, 6, ? | 2 (÷3) |
| n² ± k | Terms sit 1 above or below squares | 2, 5, 10, 17, 26, ? | 37 (n² + 1, n = 6) |
| n³ ± k | Terms sit near cubes | 0, 7, 26, 63, 124, ? | 215 (n³ − 1, n = 6) |
| Products n × (n + k) | Terms factorise neatly | 3, 8, 15, 24, 35, ? | 48 (6 × 8; also 7² − 1) |
| Squares of primes | Terms are 4, 9, 25, 49... | 4, 9, 25, 49, 121, ? | 169 (13²) |
| Sum of previous two | Each term = previous two added | 1, 4, 5, 9, 14, 23, ? | 37 (14 + 23) |
| Two interleaved series | Zig-zag; long series | 2, 20, 4, 17, 8, 14, 16, ? | 11 (odd places ×2; even places −3) |
| Mixed ×a + b | Each term ~ previous × a | 2, 4, 10, 28, 82, ? | 244 (×3 − 2) |
| Alternating operations | Two rules take turns | 5, 10, 13, 26, 29, ? | 58 (×2, +3, ×2, +3, ×2) |
Wrong-number-out: the method
- Find a rule that works for most consecutive pairs (usually the first three or four terms).
- Generate the full correct series from the first term using that rule.
- Compare term by term. The single term that differs is the answer.
- Confirm: once you fix that term, the next term should follow from the corrected value.
Example: 6, 10, 18, 35, 66, 130. Rule ×2 − 2 gives 6, 10, 18, 34, 66, 130. So 35 is wrong (should be 34), and 34 × 2 − 2 = 66 confirms it.
Squares 1–30
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 11 | 121 | 21 | 441 |
| 2 | 4 | 12 | 144 | 22 | 484 |
| 3 | 9 | 13 | 169 | 23 | 529 |
| 4 | 16 | 14 | 196 | 24 | 576 |
| 5 | 25 | 15 | 225 | 25 | 625 |
| 6 | 36 | 16 | 256 | 26 | 676 |
| 7 | 49 | 17 | 289 | 27 | 729 |
| 8 | 64 | 18 | 324 | 28 | 784 |
| 9 | 81 | 19 | 361 | 29 | 841 |
| 10 | 100 | 20 | 400 | 30 | 900 |
Cubes 1–15
| n | n³ | n | n³ | n | n³ |
|---|---|---|---|---|---|
| 1 | 1 | 6 | 216 | 11 | 1331 |
| 2 | 8 | 7 | 343 | 12 | 1728 |
| 3 | 27 | 8 | 512 | 13 | 2197 |
| 4 | 64 | 9 | 729 | 14 | 2744 |
| 5 | 125 | 10 | 1000 | 15 | 3375 |
Primes up to 100 (25 primes)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Speed tips
- Write differences in a row directly below the series; do not do them in your head.
- Use the options: if only one option fits the parity (odd or even) or last digit your rule predicts, mark it and move on.
- When terms are 1 less or 1 more than squares (35, 48, 63 or 37, 50, 65), think n² − 1 or n² + 1 straight away.
- Numbers like 7, 26, 63, 124 or 9, 28, 65, 126 are cubes − 1 or cubes + 1.
- For a series of 7 or more terms, test interleaving before anything complicated.
- If nothing works in 45 seconds, skip and come back; series questions are meant to be quick.
Common traps
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