Number Sequences

Number sequences appear in Nature all over the place, from sunflowers to conch shells. They can also crop up either in Maths or Verbal Reasoning, and both are essential parts of 11+ and other school examinations. The trick is to be able to recognise the most common sequences and, if you find a different one, to work out the pattern so that you can find the missing values.

Common sequences

Here are a few of the commonest number sequences. For each one, I've given the rule for working out the nth value, where n stands for its position in the sequence.

Even numbers: 2, 4, 6, 8 etc. Rule: 2n

Odd numbers: 1, 3, 5, 7 etc. Rule: 2n - 1

Powers of 2: 2, 4, 8, 16 etc. Rule: 2?

Prime numbers: 2, 3, 5, 7 etc. Rule: n/a

Square numbers: 1, 4, 9, 16 etc. Rule: n²

Fibonacci sequence: 1, 1, 2, 3 etc. Rule: n/a (each successive number is produced by adding the previous two numbers together, eg 1 + 1 = 2, 1 + 2 = 3)

Working out the pattern

The best way to approach an unfamiliar sequence is to calculate the gaps between the values. Most sequences involve adding or subtracting a specific number, eg 4 in the case of 5, 9, 13, 17 etc. Sometimes, the difference will rise or fall, as in 1, 2, 4, 7 etc. If you draw a loop between each pair of numbers and write down the gaps (eg +1 or -2), the pattern should become obvious, enabling you to work out the missing values. If the missing values are in the middle of the sequence, you can still work out the pattern by using whatever values lie next to each other, eg 1, ., 5, 7, ., 11 etc. You can confirm it by checking that the gap between every other value is double that between the ones next to each other. If the gaps between values are not the same and don't go up (or down) by one each time, it may be that you have to multiply or divide each value by a certain number to find the next one, eg 16, 8, 4, 2 etc. If the gaps go up and down, there may be two sequences mixed together, which means you'll have to look at every other value to spot the pattern, eg 1, 10, 2, 8 etc. Here, every odd value goes up by 1 and every even value falls by two.

Generating a formula

At more advanced levels, you may be asked to provide the formula for a number sequence with constant gaps. This is ??n ± k, where ?? is the gap, n is the position of the value in the sequence and k is a constant that is added or subtracted to make sure the sequence starts with the right number, eg the formula for 5, 8, 11, 14 etc is 3n + 2. The gap between each value is 3, which means you have to multiply n by 3 each time and add 2 to get the right value, eg for the first value, n = 1, so 3n would be 3, but it should be 5, so you have to add 2 to it. Working out the formula for a sequence is particularly useful at 13+ or GCSE level, when you might be given a drawing of the first few patterns in a sequence and asked to predict, say, the number of squares in the 50th pattern. You can also work out the position of the pattern in the sequence if you are given the number of elements. You do this by rearranging the formula, ie by adding or subtracting k to the number of elements and dividing by ??. For example, if 3n +2 is the formula for the number of squares in a tiling pattern, and you have 50 squares in a particular pattern, the number of that pattern in the sequence can be given by subtracting 2 and dividing by 3, giving the answer 16.