How to solve number sequences

Number sequences appear in all sorts of logic tests. You are given a sequence of numbers and your task is to fill in the number that should logically come next. In this article we will show how to solve such sequences.

The task

A classic number sequence question looks like this:

$$2,4,6,8,10,?$$

Your task is to find the number that can replace the question mark so that it makes the most sense. You have to find some characteristic property linking the given numbers and use it to work out the missing one. In this case the answer is 12, because each number in the sequence is always 2 more than the previous one.

The answer is not always unique. You could say that any number can replace the question mark, as long as you can justify your choice.

Adding and multiplying

Among the simplest sequences are those where you just add a number. Examples:

$$3,10,17,24,?$$

Here we always add 7, so the number in place of the question mark will be 31. Another example:

$$-8,-5,-2,1,?$$

We always add 3, so the next number in the sequence will be 4. Another example:

$$17,8,-1,-10,?$$

In this sequence we add −9, in other words we subtract 9. The next one is −19. Another example:

$$2,4,8,16,?$$

Here we are no longer adding but multiplying. Each number is twice the previous one, so the answer will be 32. Another example:

$$27,9,3,1,?$$

In this sequence we divide instead, by three. The answer will be $\frac{1}{3}$. We can also add a number that is not constant:

$$2,4,7,9,12,?$$

We alternate between adding +2 and +3. So we add +2 to 2 and get 4. Then we add +3 and get 7. And so on. We add 2 to 12 and get 14. Another example:

$$5,6,8,11,15,20,?$$

In this sequence we add a number one higher than in the previous step, starting with +1. So 5 + 1 = 6, 6 + 2 = 8, 8 + 3 = 11 and so on. Finally we add 6, so the answer is 20 + 6 = 26. Another example:

$$1,2,6,12,36,72,?$$

We alternate between multiplying by two and by three. 1 · 2 = 2, 2 · 3 = 6, 6 · 2 = 12 and so on. Finally we multiply by three, so the answer is 216.

Alternating sequences

We can combine sequences and operations further. For example, we can use addition once and multiplication the next time:

$$3,12,14,56,58,232,?$$

Here we first multiply by four and in the next step add two. So: 3 · 4 = 12, 12 + 2 = 14, 14 · 4 = 56, and so on. Finally we add two, so the answer is 234. A similar example:

$$3,9,5,15,11,33,29,?$$

First we multiply by three, then subtract four. So: 3 · 3 = 9, 9 − 4 = 5, 5 · 3 = 15 and so on. Finally we multiply by three, and the answer is 87.

We can combine sequences so that one sequence actually contains two, each with its own characteristic property. Example:

$$2,1,4,6,8,11,16,16,32,?$$

In the odd positions we have the familiar sequence 2, 4, 8, 16, 32, where each number is double the previous one, and in the even positions we have the sequence 1, 6, 11, 16, ?. This is a sequence where we always just add five. The answer is therefore 16 + 5 = 21. One more example:

$$2,7,3,11,5,15,8,19,12,?$$

In this sequence we can find two subsequences. In the odd positions we have 2, 3, 5, 8, 12, a sequence where we always add a number one higher than in the previous step. And in the even positions we have 7, 11, 15, 19, ?. Here we simply add four. The answer is 23.

Classic sequences

There are some classic sequences worth mentioning. Let’s start:

• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, … – the sequence of squares of the positive integers. Written another way: 1², 2², 3², 4², 5², … If you see any of these numbers in a sequence, especially the larger ones, there is a good chance that squaring plays a part.

• 0, 1, 1, 2, 3, 5, 8, 13, 21, … – this is the famous Fibonacci sequence. Each number is defined as the sum of the two previous numbers. So, going backwards: 21 = 8 + 13, 13 = 5 + 8, 8 = 3 + 5 and so on. The first two numbers, zero and one, are fixed.

• 2, 3, 5, 7, 11, 13, 17, 19, … – the sequence of prime numbers. Just be careful: one is not a prime number.

• 1, 2, 6, 24, 120, 720, 5040, 40320, … – the sequence of factorials. The number in the nth position is given by the product 1 · 2 · 3 · … · n. For example, the number in the fourth position is 24, which is 1 · 2 · 3 · 4 = 24.

Other types of sequences

Example:

$$17,3,6,14,-3,23,19,?$$

The solution is that if we add the numbers in the first and second places, we get twenty: 17 + 3 = 20. The same holds for the following pairs: 6 + 14 = 20, −3 + 23 = 20, 19+? = 20. So the question mark should be replaced by one. Another example:

$$?,7,12,4,5,9,6,-1,5$$

In this sequence we split the numbers into triples a, b, c, where a + b = c. So ?+7 = 12 and 4 + 5 = 9. The question mark should therefore be replaced by 5. Another example:

$$2,4,10,28,?$$

Here we have combined two operations: multiplying and subtracting. We get the next number by multiplying the previous one by three and then subtracting two. So we have: 4 = 2 · 3 − 2, 10 = 4 · 3 − 2, 28 = 10 · 3 − 2. The next one in the sequence is 28 · 3 − 2 = 82.