## Number Sequences

In the sequence 2, 4, 6, 8, 10... there is an obvious pattern. Such sequences can be expressed in terms of the nth term of the sequence. In this case, the nth term = 2n. To find the 1st term, put n = 1 into the formula, to find the 4th term, replace the n's by 4's: 4th term = 2 × 4 = 8.

Example

What is the nth term of the sequence 2, 5, 10, 17, 26... ?

To find the answer, we experiment by considering some possibilities for the nth term and seeing how far away we are:

n = 1 2 3 4 5

n² = 1 4 9 16 25

n²+1 = 2 5 10 17 26

This is the required sequence, so the nth term is n² + 1. There is no easy way of working out the nth term of a sequence, other than to try different possibilities.

Tips: if the sequence is going up in threes (e.g. 3, 6, 9, 12...), there will probably be a three in the formula, etc. In many cases, square numbers will come up, so try squaring n, as above. Also, the triangular numbers formula often comes up. This is n(n + 1)/2 .

Example

Find the nth term of the sequence: 2, 6, 12, 20, 30...

n = 1 2 3 4 5

n(n + 1)/2 = 1 3 6 10 15

Clearly the required sequence is double the one we have found the nth term for, therefore the nth term of the required sequence is 2n(n+1)/2 = n(n + 1).

The Fibonacci sequence

The Fibonacci sequence is an important sequence which is as follows: 1, 1, 2, 3, 5, 8, 13, 21, ... . The next term of this well-known sequence is found by adding together the two previous terms.