How to find the nth term of a number sequence (idea)

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 + n² - 2n - n + 2<

> 3 + n² - n

Proof, by finding the 4^th term:

3 + n² - n
3 + 16 - 4

= 15.