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 , where means any position in the sequence.

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 |

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

Position | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Operation | ||||

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 , then the position to term rule is .

The nth term of a sequence is the position to term rule using to represent the position number.

Work out the nth 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 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 | ||||

2 times table | 2 | 4 | 6 | 8 |

Operation | ||||

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 , then this is which can be written as .