Sequence tests reward a search procedure, not inspiration. Run the checklist in order.
1. First differences
Subtract consecutive terms. Constant difference means arithmetic.
3, 7, 11, 15 has differences of 4. Next: 19.
2. Ratios
Divide consecutive terms. Constant ratio means geometric.
3, 6, 12, 24 doubles. Next: 48.
3. Second differences
If first differences are not constant, difference them again. Constant second differences mean a quadratic.
2, 6, 12, 20 has differences 4, 6, 8 and second differences of 2. Next difference 10, so 30. (These are n(n+1).)
4. Interleaving
The most common trick in harder tests. Two sequences alternate.
1, 10, 2, 20, 3, 30 splits into 1, 2, 3 and 10, 20, 30. Next: 4.
Whenever a sequence looks erratic, check odd and even positions separately before anything else.
5. Recursive rules
Each term from the previous ones.
1, 1, 2, 3, 5, 8 is Fibonacci. Also check a(n) = 2a(n-1) + 1 and similar forms.
6. Known sequences
Learn to recognise on sight:
- Squares: 1, 4, 9, 16, 25
- Cubes: 1, 8, 27, 64
- Primes: 2, 3, 5, 7, 11, 13
- Triangular: 1, 3, 6, 10, 15
- Powers of 2: 1, 2, 4, 8, 16
- Factorials: 1, 2, 6, 24, 120
7. Digit-based rules
When arithmetic fails entirely, look at the digits themselves - digit sums, reversals, counts, or the look-and-say sequence (1, 11, 21, 1211, 111221).
8. Position-dependent operations
The operation itself changes with position: plus 1, times 2, plus 3, times 4.
Under time pressure
Do not stare. Run the checklist. Differences and ratios take five seconds and resolve most items; interleaving resolves most of the rest. If nothing lands within your time budget, move on and come back - on a penalty-scored test, guessing is negative EV.
Practise with the sequences drill and see sequence and pattern recognition tests.