Number Series: Spotting the Pattern Quickly

Number series questions ask for the next number in a sequence. There are only a handful of common patterns, and a routine helps you find them.

A routine to follow

1. Look at the differences between consecutive terms.
2. If the differences aren't constant, look at the differences of the differences.
3. Try ratios (multiply or divide).
4. Look for squares, cubes and primes.
5. Check for alternating patterns in odd and even positions.

Common patterns

Pattern	Example	Next term
Arithmetic (add a constant)	4, 9, 14, 19, 24	29
Geometric (multiply)	3, 6, 12, 24	48
Squares	1, 4, 9, 16, 25	36
Cubes	1, 8, 27, 64	125
Primes	2, 3, 5, 7, 11	13
Fibonacci (sum of previous two)	1, 1, 2, 3, 5, 8	13
Increasing differences	2, 6, 12, 20, 30	42

Worked example: increasing differences

2, 6, 12, 20, 30, ?

• Differences: 4, 6, 8, 10.
• These increase by 2, so the next difference is 12.
• Next term = 30 + 12 = 42.

Notice that the terms are n(n + 1): 1×2, 2×3, 3×4, 4×5, 5×6, 6×7 = 42.

Alternating series

Two series interleaved: 2, 10, 4, 20, 6, 30, ?

• Odd positions: 2, 4, 6, so next is 8.
• Even positions: 10, 20, 30.
• The next term is the seventh, which is 8.

Mixed operations

Example: 3, 7, 15, 31, ?

• Each term = previous × 2 + 1: 3×2+1 = 7, 7×2+1 = 15, 15×2+1 = 31.
• Next = 31×2+1 = 63.

Finding the odd one out

Some questions ask you to identify a wrong term. Find the rule using most of the numbers, then see which one breaks it.

Tips

• Write differences below the series.
• Learn squares, cubes and primes to spot them quickly.
• Don't spend too long. If no pattern appears in a minute, move on.

Practice

The more series you see, the faster you recognise patterns.

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