## Finding the Rules or nth Terms for Simple Sequences

An arithmetic sequence is a list of numbers such as 2, 6, 20, 14, 18. To find the next term we add 4 to the last term. Our problem is to find a formula that generates the sequence. For the sequence above, to find the next term we could write

However we don't want to write down the first twenty terms to find the twenty first term, which using the above formula would mean we have to do. We need a formula called the rule or the nth term to find the twenty first term directly: Give n=21,we can findFor the case above the formula would be,

then

We find the formula forin this way

2 | 6 | 10 | 14 | 18 |

4 | 4 | 4 | 4 |

The common difference=4 so we can deduce it is asequence.

Next we construct a 4 times table and compare it with the original sequence.

4 | 8 | 12 | 16 | 20 |

2 | 6 | 10 | 14 | 18 |

To find the original sequence from the 4 times table we subtract 2, hence the rule or n ^{th term is}

Example: Find the rule for the sequence: 5, 8, 11, 14, 17

5 | 8 | 11 | 14 | 17 |

3 | 3 | 3 | 3 | 3 |

The common difference is 3 so we know it is asequence

Construct the 3 times table and and the difference between it and the original sequence

3 | 6 | 9 | 12 | 15 |

5 | 8 | 11 | 14 | 17 |

The difference is +2 so the n ^{th term or rule is}

Example: Find the rule for the sequence: 5, 3, 1, -1, -3

5 | 3 | 11 | -1 | -3 |

-2 | -2 | -2 | -2 | -2 |

The common difference is -2 so we know it is asequence

Construct the -2 times table and find the difference between it and the original seque

-2 | -4 | -5 | -8 | -10 |

5 | 3 | 1 | -1 | -3 |

The difference is +7 so the n ^{th term or rule is}