# Sequences

## In a nutshell

Sequences can be described in different ways. The term-to-term rule indicates the pattern e.g. $+2$ each time, or $\times 5$ each time. If this sequence is given, you should be able to find the rule. Given a rule, you should be able to generate the sequence. There are different types of sequences which are categorised according to the type of rule they follow.

## Types of sequences

Sequences can be categorised according to the type of rule they follow. Here are the main categories of sequences.

**NAME OF SEQUENCE** | **TERM TO TERM RULE** | **EXAMPLE** |

Arithmetic/Linear | $+$ or $-$ each time, the $1st$ difference is the same. The $1st$ difference is the difference between adjacent terms. | $1, 6, 11, 16, 21$ |

Geometric | $\times$ or $\div$ each time. The ratio between adjacent terms is always the same. | $3, 6, 12, 24, 48$ |

Periodic | There is a repeated pattern or section. | $1, 6, 3, 1, 6, 3, 1$ |

Fibonacci | Each term is the sum of the two previous terms. | $1, 1, 2, 3, 5, 8, 13$ |

Quadratic | $n^2$ is in the nth term formula. The $2nd$ differences are the same. | $2, 5, 10, 17, 26$ |

## Generating sequences

To generate a sequence, start with the first term and then follow the rule.

##### Example 1

*Generate the first 5 terms of the sequence with the term-to-term rule*

*First term* $=4$, *Rule* $+3$ *each time.*

$\underline{4, 7, 10, 13, 16}$

##### Example 2

*Generate the first 5 terms of the sequence with the term-to-term rule*

*First term* $=-2$, *Rule* $\times -2$ *each time.*

$\underline{-2, 4, -8, 16, -32}$

## Find the rule from a sequence

To find the rule, compare adjacent terms in the sequence. Make sure the rule works for all numbers in the sequence.

If the difference between terms is the same, then the rule would be arithmetic, so work out what number to add or subtract each time. If adjacent terms are divided and the ratio is the same, then the rule would be geometric, so work out what number to multiply or divide by each time. If the numbers oscillate from positive to negative, it usually means multiplying by a negative number.

##### Examples

**SEQUENCE** | **RULE** |

$1, 6, 11, 16, 21$ | $+5$ *each time* |

$3, 6, 12, 24, 48$ | $\times 2$ *each time* |

$5, -5, 5, -5, 5$ | $\times (-1)$ *each time* |