Number Series

A working order

Do these in order and stop at the first that works:

1. First differences. Constant → arithmetic. Growing → take the second differences; constant there means a quadratic pattern (often $n^2$ based).
2. Ratios. Constant → geometric. Nearly constant → multiply-and-add.
3. Squares, cubes and their neighbours: $n^2$, $n^2\pm 1$, $n^3$, $n^3\pm n$.
4. Primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) or their squares.
5. Alternating: two patterns interleaved — look at terms 1, 3, 5 and 2, 4, 6 separately.
6. Mixed operation: $\times 2+1$, $\times 3-2$, $+1\times 2$ repeated.
7. Digit rules: sum of the digits added, digits reversed, digits multiplied.

The three question types

In a wrong term question the answer is the term to be removed, not its correct replacement — read what the options represent.

Patterns worth recognising on sight

• $2,6,12,20,30\dots$ → $n(n+1)$
• $1,4,9,16,25\dots$ → $n^2$; $2,5,10,17,26\dots$ → $n^2+1$
• $1,8,27,64\dots$ → $n^3$; $0,7,26,63\dots$ → $n^3-1$
• $1,1,2,3,5,8\dots$ → Fibonacci (each term is the sum of the two before)
• $1,2,6,24,120\dots$ → factorials
• $4,9,25,49,121\dots$ → squares of primes
• Differences that are themselves $2,4,6,8$ or $1,2,3,4$ → a quadratic rule