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 wellknown
sequence is found by adding together the two previous terms.
Copyright © Matthew Pinkney 2003
