The

n^{th} term of a

number sequence is a

formula that gives you the number at position n in that sequence. There are two different formulae for calculating the nth term, and which one you use depends on the sequence.

### The common difference formula

Imagine the sequence: 2, 4, 6, 8, 10, ... - We want to work out the nth term for this sequence. The formula used here is

`dn + ( a - d )`

Where -

`d` = Common difference between all terms.

`a` = First term.

An example of its use on this sequence -

`dn + ( a - d )`

2n + ( 2 - 2 )

`
n`^{th} term = 2n

Proof, by finding the third term -

`2n`

2 * 3

`= 6
`

### Changing difference formula

This is a bit more complicated, and is applied to sequences where the difference between each number is not a constant, as in the sequence: 3, 5, 9, 15, 24, ... The trick here is to find the

difference increase - so, the difference between 3 and 5 is 2, the difference between 5 and 9 is 4, the difference between 9 and 15 is 6... see the

pattern emerging? The difference increase is

**2**. So, we use the formula -

`a + (n-1)d + 0.5(n-1)(n-2)C`

Where -

`d` = the first difference (2 in the sequence above).

`C` = The difference increase.

`a` = the first term.

So, an example of its use, using the sequence above -

`a + (n-1)d + 0.5(n-1)(n-2)C`

3 + (n-1)2 + 0.5(n-1)(n-2)2

3 + 2n - 2 + n^{2} - 2n - n + 2<

`>
3 + n`^{2} - n

Proof, by finding the 4^{th} term:

`3 + n`^{2} - n

3 + 16 - 4

`
= 15.`