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)​