A number sequence is a list of numbers that follows a rule, so that each number can be worked out from the ones before it. Each number in the list is called a term. In 3, 7, 11, 15, for example, the rule is “add 4”, so the next term is 19. In aptitude tests you are shown part of a sequence and asked to find the next or missing term.
Number sequence questions appear in many employer tests, and you can try some in a free numerical reasoning test with a worked solution for every answer.
What a number sequence means
The word sequence simply means an ordered list. What turns a list of numbers into a number sequence is the rule linking them. The rule might be adding the same amount each time, multiplying by the same amount, or something less obvious, such as two rules taking turns.
You will also see the terms number series and number pattern. In tests they mean the same thing: find the rule, then apply it.
A typical question shows four to seven terms with one or two gaps, and offers four or five answer options. The arithmetic is usually simple. The difficulty is spotting the rule quickly.
The main types of number sequence
Most test sequences belong to a small number of families. Knowing them turns guesswork into recognition.
Arithmetic sequences
The same amount is added or subtracted each time.
1, 5, 9, 13, 17, ?
Each term is 4 more than the one before, so the next term is 21.
Geometric sequences
Each term is multiplied or divided by the same amount. Terms do not have to be whole numbers.
120, 60, 30, ?, 7.5, 3.75
Each term is half the one before. The missing term is 30 ÷ 2 = 15, and 15 ÷ 2 = 7.5 confirms it.
Two step sequences
Each term comes from the previous one by two operations, often a multiplication and an addition.
2, 5, 11, 23, 47, ?
The rule is “double, then add 1”: 2 × 2 + 1 = 5, 5 × 2 + 1 = 11, and so on. The next term is 47 × 2 + 1 = 95.
Alternating sequences
Two separate sequences are interleaved, so each term relates to the one two places back, not the one next to it.
7, 18, 14, 16, 28, 14, ?, ?
Take the 1st, 3rd and 5th terms: 7, 14, 28. Each doubles. Take the 2nd, 4th and 6th: 18, 16, 14. Each falls by 2. So the next two terms are 56 and 12.
Fibonacci style sequences
Each term is the sum of the two before it.
4, 4, 8, 12, ?, 32, 52
4 + 4 = 8 and 4 + 8 = 12, so the missing term is 8 + 12 = 20. Check: 12 + 20 = 32 and 20 + 32 = 52.
Square, cube and prime sequences
Some sequences use special numbers. 1, 4, 9, 16, 25 are the square numbers, so the next is 6 × 6 = 36. 1, 8, 27, 64 are the cubes, so the next is 5 × 5 × 5 = 125. And 2, 3, 5, 7, 11, 13 are the primes, so the next is 17.
How to find the rule in a number sequence
Work through these checks in order. Most sequences give way by the second or third.
- Look at the direction and size. Are the terms rising, falling or both? Are they growing steadily or speeding up? Steady growth suggests adding. Rapid growth suggests multiplying.
- Write the differences between terms. If the differences are all the same, it is arithmetic.
- If the differences are not constant, look at them as a sequence. Differences that rise steadily, or that double, point to the rule.
- Try dividing each term by the one before. A constant result means a geometric sequence.
- Check alternate terms. If nothing fits, split the sequence into odd and even positions.
- Look for squares, cubes, primes or sums of earlier terms.
- Test your rule on every term. A rule that fits the first three terms but not the fifth is wrong.
Worked example
3, 5, 9, 15, 23, ?
The terms rise, but not by a fixed amount. Write the differences: 2, 4, 6, 8. The differences go up by 2 each time, so the next difference is 10. The next term is 23 + 10 = 33.
A second worked example
6, 11, 21, 41, 81, ?
The differences are 5, 10, 20, 40. They double each time, so the next difference is 80 and the next term is 81 + 80 = 161. You can also see the rule directly: each term is double the previous one, minus 1. 81 × 2 − 1 = 161. When two routes give the same answer, you can be confident in it.
Tips for number sequence questions
- Never assume from the first few terms. A pattern that looks obvious early can change later on. Check the whole sequence.
- Work backwards when stuck. Starting from the last term can make a halving or subtracting rule easier to see.
- Use the answer options. If only one option is even, or only one is close to the size you expect, that narrows things quickly.
- Watch for gaps in the middle. The method is the same: find the rule from the terms you have, then fill the gap and check it works on both sides.
- Keep moving. If a sequence has not given way after a minute, make your best choice and come back if time allows.
Where number sequences appear
Number sequence questions appear in numerical reasoning tests, basic numeracy tests and some general aptitude tests. They check pattern spotting and comfort with arithmetic rather than advanced maths. For a broader picture of the skills employers test, see our guides to what numerical reasoning is and what numeracy skills are, or look through more numerical reasoning test examples.
The fastest way to improve is regular timed practice. Try a free numerical reasoning test and read the explanation for any sequence you miss, so the next one of that type feels familiar.
