The differences ladder (1st differences → 2nd differences) is the master key. If the first differences aren't constant, check the second differences before guessing multiplication.

Pattern Catalogue

1. Arithmetic — constant difference: $a_n = a_1 + (n-1)d$.
2. Geometric — constant ratio: $a_n = a_1\,r^{\,n-1}$.
3. Quadratic — 1st differences form an AP (2nd differences constant).
4. Cubic — needs a third layer of differencing.
5. Square-based — terms are $n^{2} \pm$ constant (e.g. 3, 8, 15, 24, 35 $= n^{2}+2n$).
6. Cube-based — terms are $n^{3} \pm$ constant (e.g. 9, 28, 65, 126 $= n^{3}+1$).
7. Primes — 2, 3, 5, 7, 11, … or a twist (alternate primes, primes ±1).
8. Alternating operations — e.g. ×3, −1, ×3, −1 (2, 6, 5, 15, 14, 42).
9. Interleaved — odd and even positions follow separate rules (split first).
10. Multiplication / division — a changing factor (2, 6, 24, 120 → ×3, ×4, ×5).
11. Fibonacci-style — each term = sum of the previous two.
12. Digit-sum — the next term uses the digits of the current one (12 → 12+1+2 = 15…).
13. Wrong number / odd-one-out — find the majority rule, then the term that breaks it.

Solving Checklist (in order)

1. Simple AP or GP? 2. First differences constant / their own AP? 3. Squares, cubes, primes near each term? 4. Split odd / even positions? 5. Alternating operations? 6. Digit manipulation.

Speed trap: never lock a rule after two terms — verify it against three consecutive gaps.