Position to term rules

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 for arithmetic sequences

Example

Work out the position to term rule for the following sequence: 5, 6, 7, 8, ...

First, write out the sequence and the positions of each term.

Position	1	2	3	4
Term	5	6	7	8

Position
1
2
3
4

Term
5
6
7
8

Next, work out how to go from the position to the term.

Position	1	2	3	4
Operation	\(+4\)	\(+4\)	\(+4\)	\(+4\)
Term	5	6	7	8

Position
1
2
3
4

Operation
\(+4\)
\(+4\)
\(+4\)
\(+4\)

Term
5
6
7
8

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 \(nth\) term of the following sequence: 3, 5, 7, 9, ...

Firstly, write out the sequence and the positions of the terms.

Number positions of a sequence with +2 increments

As the rule for going from the position to the term is not obvious, look for the differences between the terms. In this case, there is a difference of 2 each time.

This difference describes the times tables that the sequence is working in. In this sequence it is the 2 times tables.

Write out the 2 times tables and compare each term in the sequence to the 2 times tables.

Position	1	2	3	4
Operation	\(\times 2\)	\(\times 2\)	\(\times 2\)	\(\times 2\)
2 times table	2	4	6	8
Operation	\(+ 1\)	\(+ 1\)	\(+ 1\)	\(+ 1\)
Term	3	5	7	9

Position
1
2
3
4

Operation
\(\times 2\)
\(\times 2\)
\(\times 2\)
\(\times 2\)

2 times table
2
4
6
8

Operation
\(+ 1\)
\(+ 1\)
\(+ 1\)
\(+ 1\)

Term
3
5
7
9

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 \(2n + 1\).

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Sequences - AQAPosition to term rules