A sequence is a set of numbers arranged in a particular order.

Sequences can be:

finite, a fixed number of terms e.g 3, 4orinfinite, going on forever e.g. 3, 4, 5, 6,...

This course studies infinite sequences.

Many sequences have a mathematical pattern. **Arithmetic** and

**Geometric**sequences are examples of sequences with a patterns and are studied in the next two topics. Sequences can be enclosed in triangular brackets. <3, 4, 5...>

### Notation

Each number in a sequence is called a **term.** There are various ways of describing terms.

The first term of a sequence can be given by t_{1}, t(1), T(1) or T_{1}. We will use t_{1}.

The **general **term or **n **th term is written as t_{ n}.

There are four ways to express a sequence.

Method |
Explanation |
Example |

By listing |
List the first four or five terms in order. | 2, 8, 23, 68... |

By formula for the nth term |
A formula is provided to work out any term of the sequence. This activity will help you to work out the formula for sequences. |
<3n + 2> or t _{1} = 3 x 1 + 2 = 5Second term, t _{2} = 3 x 2 + 2 = 8Third term t _{3} = 3 x 3 + 2 = 11 etc.
The sequence is 5, 8, 11... |

By a recursive function |
One term is provided and a formula is given to work out the following term. |
t First term is 3 The sequence is 3, 7, 15... |

By graphing |
The sequence is illustrated on a graph. The The |
Graph < 1, 3, 5, 7... > |