Number series: common patterns and how to spot them

A number series shows five or six numbers and asks for the next one. Every series follows a rule, and the rule always uses simple arithmetic. The challenge is finding it. Here are the patterns you will meet, from easiest to hardest, with original examples.

The first move: look at the gaps

Almost every series gives itself away through the differences between neighbouring numbers. Take this one:

7, 11, 15, 19, 23, ?

The gaps are +4, +4, +4, +4. The next number is 27.

Whenever you are stuck, write the gaps down. Then, if needed, write the gaps between the gaps.

Common patterns

Constant step

The same number is added or subtracted each time.

• 12, 19, 26, 33, 40 → 47 (add 7)
• 88, 81, 74, 67, 60 → 53 (subtract 7)

Constant ratio

Each number is multiplied by the same factor.

• 3, 6, 12, 24, 48 → 96 (×2)
• 2, 6, 18, 54, 162 → 486 (×3)

A quick sign: the gaps themselves grow fast and double or triple each time.

Growing gaps

The gap changes by a fixed amount each step.

• 4, 5, 7, 10, 14, 19 → 25 (gaps +1, +2, +3, +4, +5, so next +6)

Squares plus a constant

• 6, 11, 18, 27, 38 → 51 (these are 2², 3², 4², 5², 6² plus 2, so 7² + 2)

Gaps that rise by 2 each time (5, 7, 9, 11…) are a strong hint that squares are involved.

Two operations taking turns

• 3, 8, 16, 21, 42, 47 → 94 (add 5, then double, and repeat)

The gaps look irregular, but every second step is a multiplication.

Two series woven together

• 2, 30, 5, 27, 8, 24 → 11

Read every second number: 2, 5, 8 rises by 3, while 30, 27, 24 falls by 3. The next number belongs to the first series.

Sum of the previous two

• 2, 3, 5, 8, 13, 21 → 34

Each number is the sum of the two before it. A variant adds a small extra amount each time.

Alternating gaps

• 5, 7, 12, 16, 19, 25 → 26

The gaps are +2, +5, +4, +3, +6. Split them: +2, +4, +6 rise by 2, while +5, +3 fall by 2, so the next gap is +1.

Multiply, then adjust

• 2, 5, 11, 23, 47 → 95 (double and add 1 each time)

A method for hard series

1. Write the gaps. Constant? Done.
2. Check ratios if the numbers grow quickly.
3. Write the gaps between the gaps if the gaps change steadily.
4. Split odd and even positions if the numbers jump up and down.
5. Check sums: is each number built from the two before it?
6. Check your rule against every number, not just the last few.

Wrong options in the test are built from typical slips: repeating the last gap when it should change, continuing the wrong one of two woven series, or an arithmetic slip of one. If your first answer matches an option instantly, double-check it fits the whole series.

Practise

Practise number series for free, with feedback and an explanation after every answer, or take the quick test to see series alongside matrices and rotation puzzles.