Number sequence questions look like they need inspiration. They do not. Almost every one in an aptitude test, an IQ-style quiz or a puzzle app belongs to one of about ten families, and a fixed checklist finds the family in under a minute. Here is the checklist in the order that finds the answer fastest.
Step 1: Write down the gaps
Subtract each term from the next. This one step settles most questions.
- Gaps all the same (2, 5, 8, 11 → gaps 3, 3, 3): an arithmetic sequence. Add the gap. Next is 14.
- Gaps growing by the same amount (2, 5, 11, 20 → gaps 3, 6, 9): a quadratic sequence. The next gap is 12, so the next term is 32.
- Gaps doubling or tripling (5, 7, 11, 19 → gaps 2, 4, 8): the gaps are geometric. Next gap is 16, next term 35.
Step 2: If gaps fail, try ratios
Divide each term by the one before. 3, 6, 12, 24 gives 2, 2, 2: a geometric sequence, next is 48. 81, 27, 9, 3 gives ⅓ each time, so next is 1. If the ratios are close to constant but not exact, look for “multiply then add”: 2, 5, 11, 23 is ×2 + 1 each time, next 47.
Step 3: Check the famous families
Some sequences are recognisable on sight once you have seen them a few times. Memorise the first eight terms of each.
| Family | Terms | Tell |
|---|---|---|
| Squares | 1, 4, 9, 16, 25, 36, 49, 64 | Gaps are the odd numbers |
| Cubes | 1, 8, 27, 64, 125, 216 | Grows fast; second gaps rise by 6 |
| Triangular | 1, 3, 6, 10, 15, 21, 28 | Gaps are 2, 3, 4, 5 … |
| Fibonacci | 1, 1, 2, 3, 5, 8, 13, 21 | Each term is the sum of the previous two |
| Primes | 2, 3, 5, 7, 11, 13, 17, 19 | Irregular gaps, all odd after 2 |
| Powers of 2 | 1, 2, 4, 8, 16, 32, 64 | Doubling |
Step 4: Split it in two
If nothing fits and the gaps look chaotic, the sequence is probably two sequences taking turns. Read every other term. 2, 10, 4, 20, 6, 30 is 2, 4, 6 interleaved with 10, 20, 30; next is 8. A close cousin is the alternating operation: 3, 6, 4, 8, 6, 12 is ×2, −2, ×2, −2; next is 10.
Step 5: Look at the digits themselves
Rarely, the rule is about the written numerals rather than their values. 1, 11, 21, 1211, 111221 describes the previous term aloud (“one 1”, “two 1s”, “one 2, one 1”). 18, 27, 36, 45 have digits summing to 9. If the arithmetic checks all fail on a sequence that looks too tidy to be random, check the digits.
Practise the checklist, not the answers
The aim is to run steps 1 to 4 automatically. Brisk’s brain teaser of the day gives you two sequences every morning that stay within the first three families, so you can rehearse the routine under a clock. When you meet a sequence you cannot crack, paste it into the sequence solver, which runs the same checklist and shows its working.
Common questions
How long should a sequence question take?
In a numerical reasoning test, 30 to 45 seconds. Run the checklist in order and move on if the first four checks fail; the harder patterns are rarer and you can return to them.
What if two rules both fit?
Tests are written so that the simplest rule gives the intended answer. Simplest means: constant difference before constant ratio, single sequence before interleaved, and integer operations before fractional ones.
Are letter sequences the same?
Usually. Convert letters to their position in the alphabet (A=1, B=2 …) and the same checklist applies. Watch for sequences that wrap from Z back to A.