# Sequences

## Topic Overview

A sequence is the term used to described a set of numbers which are connected by a mathematical pattern. There are a wealth of number sequences with which you will probably already be familiar, for example:

Even Numbers: 2,4,6,8,10,12,14,16,18,20....

Odd Numbers: 1,3,5,7,9,11,13,15,17,19,21.....

Powers of 2: 2, 4, 8, 16, 32, 64…

Triangular Numbers: 1, 3, 6, 10, 15, 21, 28....

The triangular number sequence is generated from a pattern of dots which form a triangle. By adding another row of dots and counting all the dots we can find the next number of the sequence. As a result, you can use the following rule to calculate terms within the triangular number sequence more quickly:

${{x}_{n}}=\frac{n(n+1)}{2}$

For example, if you wanted to find out the 21st term in the sequence, instead of adding rows upon rows of dots, you could follow this rule:

${{x}_{21}}=\frac{21(21+1)}{2}=231$

Square numbers:1, 4, 9, 16, 25, 36, 49, 64…

In the square number sequence, the next term is made by squaring where it is in the pattern. Therefore you can use the following rule:

${{x}_{n}}={{n}^{2}}$

For example, if you wanted to find out the 35th term in the sequence:

${{x}_{35}}={{n}^{2}}={{35}^{2}}=1225$

Cube numbers: 1, 8, 27, 64, 125…

In the cube number sequence, the next term is made by cubing where it is in the pattern. Therefore you can use the following rule:

${{x}_{n}}={{n}^{3}}$

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