Number series appears in every OPSC ASO reasoning paper. It is asked in three ways: find the next term, find the missing term, or find the wrong term. The last type has come every year from 2021 to 2025. Almost every series follows one of about ten patterns, so the skill is to test them in a fixed order.
Step 1: write the differences
Write the gap between each pair of terms under the series. In most questions the differences show the pattern straight away.
- Same difference → arithmetic series: 7, 12, 17, 22 (+5).
- Differences increase steadily → second-level series: 11, 14, 24, 41, 65 has differences 3, 10, 17, 24 (+7 each), so the next term is 65 + 31 = 96 (ASO 2015).
- Differences multiply → 3, 4, 6, 10, 18 has differences 1, 2, 4, 8.
- Differences shrink evenly → 325, 259, 204 … has differences 66, 55, 44, 33, 22, 11 (ASO 2021).
Step 2: if the numbers grow fast, try × and ÷
- Constant ratio: 3, 6, 12, 24 (×2). Falling: 2187, 729, 243 (÷3).
- × then + : 5, 11, 23, 47 (each = previous × 2 + 1).
- Changing multiplier: 2, 2, 4, 12, 48 (×1, ×2, ×3, ×4).
- Alternate operations: ×2, +3, ×2, +3 …
Step 3: look for squares and cubes
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
| n³ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 | 1331 | 1728 |
- n² ± k or n³ ± k: 999, 730, 511, 344, 215 = 10³−1, 9³+1, 8³−1 … (ASO 2015).
- n³ − n²: 4, 18, 48, 100 (ASO 2018).
- n(n + 1): 2, 6, 12, 20, 30, 42 (ASO 2012). For odd n only: 2, 12, 30, 56, 90 (ASO 2015).
- Squares and cubes mixed: 1, 4, 27, 16, 125, 36, 343 (ASO 2025).
Step 4: two series mixed together
If the differences jump about with no pattern, take the odd places and even places separately. 3, 5, 5, 19, 7, 41, 9, ? has odd places 3, 5, 7, 9 and even places 5, 19, 41, 71 (ASO 2018).
Other patterns
- Each term is the sum of the previous two (Fibonacci type), or their product: 2, 5, 10, 50, 500 (ASO 2025).
- Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19.
- Alternating signs in the difference: +1, −2, +3, −4 (ASO 2025).
Wrong-term questions
Find the rule from the terms that fit, then check every term against it. Usually only one term breaks the pattern. Always check the last term too, because OPSC often hides the error there (2021, 2025).