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GCSE Maths
Number
Shape and Space
Statistics and Probability
Graphs
Algebra
Trigonometry
Other
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.