# Number patterns and 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.

## Types of sequences

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

### Arithmetic/Linear

$+$ or $-$ each time, the $1st$ difference is the same. The $1st$ difference is the difference between adjacent terms.

##### Example

$1, 6, 11, 16, 21$

### Geometric

$\times$ or $\div$ each time. The ratio between adjacent terms is always the same.

##### Example

$3, 6, 12, 24, 48$

### Periodic

There is a repeated pattern or section.

##### Example

$1, 6, 3, 1, 6, 3, 1$

### Fibonacci

Each term is the sum of the two previous terms.

##### Example

$1, 1, 2, 3, 5, 8, 13$

### Quadratic

The difference between each term is not equal, but the second difference between each term is equal. These set of numbers follow a pattern based on the $n^2$ sequence (the square numbers).

##### Example

$2, 5, 10, 17, 26$

## Generating sequences

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

##### Example 1

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

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

*Answer*

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

##### Example 2

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

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

*Answer*

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

## 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

- $1, 6, 11, 16, 21$
*(Rule: *$+5$* each time)* - $3, 6, 12, 24, 48$
*(Rule: *$\times 2$* each time)* - $5, -5, 5, -5, 5$
*(Rule: $\times (-1)$ each time)*

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