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

Sequences can be:

finite, a fixed number of terms e.g 3, 4
or
infinite, 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 t1, t(1), T(1) or T1. We will use t1.
The general term or 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 tn = 3n + 2 First term, t1 = 3 x 1 + 2 = 5Second term, t2 = 3 x 2 + 2 = 8Third term t3 = 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 1 = 3 ,t n + 1 = 2 t n + 1 First term is 3Second term is 2 x 3 + 1 = 7Third term is 2 x 7 + 1 = 15        etc. The sequence is 3, 7, 15... By graphing The sequence is illustrated on a graph. The horizontal axis is the number of the term. The vertical axis is the term itself. Graph < 1, 3, 5, 7... > 