Position to term rules or nth term
Each term in a sequence has a position. The first term is in position 1, the second term is in position 2 and so on.
Position to terms rules use algebra to work out what number is in a sequence, if the position in the sequence is known. This is also called the \(n\)th term, which is a position to term rule that works out a term at position \(n\), where \(n\) means any position in the sequence.
Working out position to term rules
Example
Work out the position to term rule for the following sequence:
5, 6, 7, 8, ...
Firstly, write out the sequence and the positions of each term.
Next, work out how to go from the position to the term.
In this example, to get from the position to the term, take the position number and add 4.
If the position is \(n\), then the position to term rule is \(n + 4\).
The nth term
The \(n\)th term of a sequence is the position to term rule using \(n\) to represent the position number.
Example
Work out the \(n\)th term of the following sequence:
3, 5, 7, 9, ...
Firstly, write out the sequence and the positions of the terms.
As there isn’t a clear way of going from the position to the term, look for a common difference between the terms. In this case, there is a difference of 2 each time.
This common difference describes the times tables that the sequence is working in. In this sequence it’s the 2 times table.
Write out the 2 times table and compare each term in the sequence to the 2 times table.
To get from the position to the term, first multiply the position by 2 then add 1. If the position is \(n\), then this is \(2 \times n + 1\) which can be written as \(2n + 1\).
Question
Work out the \(n\)th term of the following sequence:
6, 13, 20, 27, …
The common difference is +7
Compare the sequence with the 7 times table:
7, 14, 21, 28, …
We need to subtract 1 from the 7 times table to get our sequence. So the \(n\)th term is \(7n - 1\).
More guides on this topic
- Equations of lines – WJEC
- Equations of curves - Intermediate & Higher tier – WJEC
- Basic algebra – WJEC
- Equations and formulae – WJEC
- Factorising - Intermediate & Higher tier – WJEC
- Quadratic expressions - Intermediate & Higher tier – WJEC
- Functions - Higher only – WJEC
- Inequalities - Intermediate & Higher tier – WJEC
- Simultaneous equations - Intermediate & Higher tier – WJEC