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 nth 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, ...

First, write out the sequence and the positions of each term. As there is a clear way of getting from the position to the term, the nth term is straightforward.

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 nth 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 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 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 - EduqasPosition to term rules or nth term