Types of Sequences

Maths: Edexcel A Level Pure Maths, topic 4.2.3

Types of Sequences

We can determine whether a sequence is increasing, decreasing or periodic by looking at patterns in the terms of the sequence.

Increasing sequence

Increasing sequence

A sequence is increasing if each term of the sequence is greater than the previous term.
We can write this condition as:
$u_{n+1}>u_n \text{ for all } n\in$ ℕ

Example

Example

These sequences are all increasing sequences:
$1, 2, 3, 4,...$
$1, 4, 9, 16, ...$
$6, 12, 24, 48, ...$

Decreasing sequences

Decreasing sequences

A sequence is decreasing if each term of the sequence is less than the previous term.
We can write this condition as:
$u_{n+1}<u_n\text{ for all }n\in$ ℕ

Examples

Examples

These sequences are all decreasing sequences:
$-4,-5,-6,-7,...$
$7, 3, 2, 0, ...$
$100, 98, 96, 94, ...$

Periodic

Periodic

A sequence is periodic if the terms of the sequence repeat after a certain number of terms.
If a sequence is periodic, then there is an integer $k$ such that:
$u_{n+k} = u_n\text{ for all }n \in$ ℕ
Where $k$ is called the order of the sequence.

Examples

Examples

The sequence $0, 1, 0, -1, 0, 1, 0, -1, ....$ is a periodic sequence of order 4, as the $n$ th term is the same as the $n+4$ th term.
The sequence $5,8,5,8,5,8,...$ is a periodic sequence of order 2, as the $n$ th term is the same as the $n+2$ th term.

None of the above

None of the above

A sequence can have terms that are not increasing, decreasing or periodic.
For example, the sequence $10, -10, 8, -8, 7, -7, ....$ is not increasing as each term is not greater than the previous term.
It is not decreasing because each term is not less than the previous term.
It is not periodic because there is no repetition of terms.

Test yourself on Types of Sequences

Which term best describes the following sequence:
$25,20,15,10,...$
Which term best describes the following sequence:
$7, 7.25, 7.5, 7.75, ...$

Practise this topic for free