Number sequences: how to solve them and the patterns that come up most
What comes next in 2, 6, 12, 20, 30… The eight patterns that cover almost every sequence in an aptitude test, with the order in which to try them.
2, 6, 12, 20, 30, …
If you have ever sat an aptitude test, an entrance exam or a recruitment assessment, you have seen this. And you have probably also watched somebody solve it in four seconds while you were still looking at the numbers, with the feeling that either you see it or you do not.
It is not like that. Number sequences are solved with a procedure, and the procedure can be learned in an afternoon. Here it is in full.
The first move, always the same
Write down the differences. Do not think about it, just do it.
2 6 12 20 30
4 6 8 10With the sequence above: the differences are 4, 6, 8, 10. You are no longer looking at a hard sequence, you are looking at an easy one going up in twos. The next difference is 12, so the answer is 30 + 12 = 42.
This alone — dropping down a level by writing the differences — solves more than half the sequences you will meet. And if the differences are not obvious either, do it again with them. Two levels are almost always enough.
The eight patterns that cover almost everything
Try them in this order, which goes from most to least common.
1. Constant addition. 3, 7, 11, 15… Equal differences. The most frequent by some margin.
2. Constant subtraction. 80, 73, 66, 59… The same but downwards. Many people dismiss a descending sequence before looking at it; do not.
3. Multiplication. 3, 6, 12, 24… If the numbers grow fast, do not waste time on differences: divide each term by the previous one. If you get the same number, that is it.
4. Division. 96, 48, 24, 12… The previous one in reverse. The tell: large numbers dropping like a stone.
5. Growing difference. 1, 2, 4, 7, 11… The differences are 1, 2, 3, 4. This is where the difference trick shines: the sequence looks like chaos and the one below it is just counting.
6. Alternating step. 5, 8, 13, 16, 21… Differences 3, 5, 3, 5. Here there are two rules taking turns. The tell: the differences are neither constant nor growing, they oscillate.
7. Squares and triangular numbers. 4, 9, 16, 25 are squares. 1, 3, 6, 10, 15 are triangular (adding 1, 2, 3, 4…). Many sequences are these two with a number added on top: 6, 11, 18, 27 is squares + 2. If the differences grow by one each time, suspect these.
8. Each term depends on the two before it. 2, 3, 5, 8, 13… Each is the sum of the previous two, the Fibonacci sequence. You recognise it because it grows fast but not by multiplication: if dividing gives no clean factor and adding the two previous terms fits, this is it.
Tricks that genuinely save time
Look at the size before anything else. It is the decision that saves the most time:
- The numbers grow slowly → differences.
- The numbers shoot up → multiplication, or the two previous terms.
- The numbers go down → subtraction or division.
Three seconds on this spares you a minute of trying what it was not.
If you are given options, use them. When the answer is multiple choice, the options are information. If the pattern you suspect gives 42 and no option is 42, do not carry on: the pattern is wrong. And the other way round, if you are torn between two rules, see which one lands on an option.
Watch out for sequences that are not numbers. Sometimes the terms are letters (A, C, F, J…) and the solution is identical: swap each letter for its position in the alphabet (1, 3, 6, 10) and you will see triangular numbers.
Rule out «no solution». Barring a printing error, every one of them has a rule. If you have spent a minute on it, it is not that the sequence is strange: it is that you are trying the wrong family. Go back to the size.
Practise recognition, not arithmetic
This is the advice that changes results most: what you train here is not calculating, it is recognising the family. Someone fast at sequences does not calculate faster than you; they rule out sooner.
And that is trained in exactly one way: by doing many, varied, and looking at the pattern after each one even when you got it right. Asking yourself «which family was that?» when you finish is worth more than doing ten more blind.
At MenteBrillante, the Logic Series puzzle is this: five terms are shown and you pick what comes next from four options. The patterns come in according to your level — it starts with addition and subtraction, and later products, growing differences, squares, Fibonacci and primes appear — so you are not hit with a Fibonacci sequence on day one.
One last thing
Number sequences have a bad reputation thanks to selection tests, where they are done against the clock and with nerves. Outside that context they are one of the cleanest puzzles there is: they do not depend on vocabulary, or general knowledge, or the language you play in. A hidden rule, and you looking for it.
They can be played free in the web version, with nothing to install. And if there is a test in front of you, we have a whole article on preparing for it.