How to crack number series questions

A step-by-step order of checks for number series in reasoning tests (differences, ratios, squares and cubes, alternating series and Fibonacci-style rules), with worked examples.

In a number series question you see a few terms and have to find the next one, or the one that doesn't fit. There are only a handful of rules that test-setters use, so the fastest approach is to check them in a fixed order instead of staring at the numbers and hoping a pattern jumps out.

Check 1: the differences

Write the difference between each pair of neighbouring terms underneath the series.

Example: 13, 16, 19, 22, 25, ?

• Differences: 3, 3, 3, 3.
• The difference is constant, so the next term is 25 + 3 = 28.

If the differences aren't constant, look at the differences of the differences.

Example: 2, 5, 10, 17, 26, ?

• Differences: 3, 5, 7, 9. These go up by 2 each time.
• The next difference is 11, so the next term is 26 + 11 = 37.

(This one is also n² + 1: 1² + 1, 2² + 1, 3² + 1… Many series can be seen in more than one way, and both give the same answer.)

Check 2: the ratios

If the terms grow quickly, the rule is probably multiplication, not addition.

Example: 3, 6, 12, 24, ?

• Each term is double the one before, so the next term is 48.

Example: 4, 9, 19, 39, ?

• Each term is (previous × 2) + 1: 4 × 2 + 1 = 9, 9 × 2 + 1 = 19, and so on.
• The next term is 39 × 2 + 1 = 79.

A useful sign: if the differences are themselves doubling (5, 10, 20), the rule involves × 2.

Check 3: squares and cubes

Keep the first squares and cubes in your head. They hide inside many series.

• Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144…
• Cubes: 1, 8, 27, 64, 125, 216, 343…

Example: 1, 8, 27, 64, ?

• These are 1³, 2³, 3³, 4³, so the next term is 5³ = 125.

Example: 0, 3, 8, 15, 24, ?

• These are 1² − 1, 2² − 1, 3² − 1… so the next term is 6² − 1 = 35.

Check 4: two series woven together

If nothing works, look at every second term. Some series alternate between two separate patterns.

Example: 2, 10, 4, 20, 6, 30, ?

• Odd positions: 2, 4, 6 (going up by 2).
• Even positions: 10, 20, 30 (going up by 10).
• The next term is in an odd position, so it is 8.

Some series alternate two operations instead, such as × 2, then + 3, then × 2, then + 3.

Check 5: each term built from the previous two

In a Fibonacci-style series, each term is the sum of the two before it.

Example: 3, 4, 7, 11, 18, ?

• 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18.
• The next term is 11 + 18 = 29.

Finding the wrong number

Sometimes the question asks which term doesn't belong. Use the same checks, but expect one number to break the pattern. Find the rule using the majority of the terms, then check each term against it.

Example: 5, 10, 20, 35, 80

• Most terms double: 5, 10, 20, then 40, 80.
• So 35 is wrong; it should be 40.

Summary: the order of checks

1. Differences (and second differences).
2. Ratios: is it multiplying?
3. Squares and cubes, plus or minus a little.
4. Alternating terms.
5. Sums of the previous two terms.

Going through these in order is quicker than guessing, and it stops you missing simple rules while looking for clever ones.

Practise

Try the solved number series examples, then letter series and mixed series, which use the same ideas with letters.

Practise it: solved examples.