How to Crack a What-Comes-Next Question

What's the next number: 2, 6, 12, 20, 30, ...?

Try to find it before reading on — and notice which technique you reach for first, because that instinct is the thing this page wants to sharpen.

A checklist, not a guess

"What comes next" puzzles feel like pattern-spotting luck, but in practice almost every one falls into a small number of buckets, checked in this order:

1. First differences. Subtract consecutive terms. If the differences are constant, it's linear (arithmetic). If the differences themselves form a recognizable pattern, look one level deeper.
2. Second differences. If first differences aren't constant but their differences are, the sequence is quadratic — a polynomial of degree 2 in the term's position.
3. Ratios. If consecutive terms have a constant ratio, it's geometric. If the ratio itself is changing predictably (e.g., ratio equals the term's index), suspect factorials or similar.
4. Known families. Squares, triangular numbers, Fibonacci-style "sum of previous terms," primes, powers of 2 — pattern-match against a short mental list before inventing something new.
5. Products of small structures. Many sequences are secretly $n(n+1)$, $n^2+1$, or similar — try writing each term as a function of its position $n = 1, 2, 3, \ldots$ directly.

Solving 2, 6, 12, 20, 30

First differences: $6-2=4$, $12-6=6$, $20-12=8$, $30-20=10$ — that's 4, 6, 8, 10, not constant, but itself an arithmetic sequence increasing by 2 each time.

Second differences: $6-4=2$, $8-6=2$, $10-8=2$ — constant. A constant second difference means the original sequence is quadratic in its position $n$.

Rather than solving a system of equations, try writing each term directly as a function of $n=1,2,3,4,5$: $2 = 1\times2$, $6=2\times3$, $12=3\times4$, $20=4\times5$, $30=5\times6$. The pattern jumps out: term $n$ equals $n(n+1)$.

The next term is $n=6$: $6 \times 7 = \textbf{42}$. Check against the difference method: the next first difference should be $10+2=12$, and $30+12=42$ — matches.

A second example: a ratio-based sequence

What comes next: 3, 6, 18, 108, ...?

First differences: 3, 12, 90 — not obviously patterned. Try ratios instead: $6/3=2$, $18/6=3$, $108/18=6$. The ratios themselves are 2, 3, 6 — not constant, so it's not simple geometric. But look at the ratios' relationship to term position: ratio from term 1→2 is 2, term 2→3 is 3, term 3→4 is 6. Try a different lens: is each ratio itself the product of the previous two ratios, or related to the terms themselves? Notice $18 = 6 \times 3$ and $108 = 18 \times 6$ — each term is the product of the previous two terms. Check: $6 \times 3 = 18$ ✓, $18 \times 6 = 108$ ✓. So the next term is $108 \times 18 = \textbf{1{,}944}$.

This is the "known family" bucket — a multiplicative Fibonacci-style recurrence — and it only surfaces once you've ruled out plain differences and plain ratios and started looking for a relationship between terms rather than within a single column of differences or ratios.

Where this generalizes

The differencing method is exactly the discrete analogue of recognizing a polynomial's degree from its derivatives, and it's the same instinct a quant uses when eyeballing a data series to guess whether it looks linear, quadratic, or exponential before fitting anything formally. The broader lesson: a "what comes next" puzzle is testing whether you have a systematic fallback (differences → ratios → known families → term-as-function-of-position) rather than whether you can spot patterns by inspiration alone — inspiration is unreliable under interview pressure; the checklist isn't.