Sequences

Linear, quadratic, exponential and Fibonacci sequences are explored

Number sequences are sets of numbers that follow a pattern or a rule.

If the rule is to add or subtract a number each time, it is called an arithmetic sequence.

If the rule is to multiply or divide by a number each time, it is called a geometric sequence.

Each number in a sequence is called a term.

A sequence which increases or decreases by the same amount each time is called a linear sequence.

Term to term rules

The term to term rule of a sequence describes how to get from one term to the next.

Example 1

Write down the term to term rule and then work out the next two terms in the following sequence.

3, 7, 11, 15, ...

Firstly, work out the difference in the terms.

Number sequence with increments of +4

This sequence is going up by four each time, so add 4 on to the last term to find the next term in the sequence.

3, 7, 11, 15, 19, 23, ...

To work out the term to term rule, give the starting number of the sequence and then describe the pattern of the numbers.

The first number is 3. The term to term rule is 'add 4'.

Once the first term and term to term rule are known, all the terms in the sequence can be found.

Example 2

Write down the term to term rule and then work out the next two terms in the following sequence.

-1, -0.5, 0, 0.5, ...

The first term is -1. The term to term rule is 'add 0.5'.

Sequence with +0.5 increments.
Question

What is the term to term rule and the next two terms of the sequence: 17, 14, 11, 8, ...?

To work out the term to term rule, give the first term and then the pattern. The first term is 17, and the pattern is to subtract 3 each time, so the term to term rule is 'start at 17 and subtract 3'.

The next two terms of the sequence are 5 and 2, giving the sequence as:

Sequence with -3 increments
Question

What are the next three terms of a sequence that has a first term of 1, where the term to term rule is multiply by 2?

The first term is given as 1. Each number that follows is double the number before.

Sequence with x2 increments