Overview
Sequence puzzles ask you to reverse-engineer a hidden rule from a few examples. The sequence 2, 6, 12, 20, 30 hides a pattern that becomes obvious the moment you look at the gaps between terms rather than the terms themselves.
This is the same skill scientists use to fit a model to data: don't stare at the numbers, look at how they change.
How to solve What Comes Next?
- Write down the first differences: 6−2=4, 12−6=6, 20−12=8, 30−20=10.
- Notice the differences are 4, 6, 8, 10 — themselves increasing by a constant 2. That is the signature of a quadratic rule.
- Extrapolate the next difference: 10 + 2 = 12.
- Add it to the last term: 30 + 12 = 42. You can also read the closed form directly: aₙ = n(n+1).
The key insight
When the first differences don't reveal the pattern, look at the differences of the differences. Constant second differences always mean a quadratic formula is at work.
Variations & echoes
- The same terms are the 'pronic' numbers n(n+1), and also twice the triangular numbers.
- The method of finite differences generalises to any polynomial sequence.
- Beware: any finite list has infinitely many 'valid' continuations — puzzles rely on the simplest one.
Frequently asked questions
Couldn't the next number be something else?
Technically yes — any number can be justified by a complicated enough rule — but 42 follows the simplest pattern, which is the convention these puzzles assume.
What's the formula?
aₙ = n(n+1): the 6th term is 6·7 = 42. Equivalently, each term adds the next even number.