The
nth 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 )
nth 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 4th term:
3 + n2 - n
3 + 16 - 4
= 15.