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Explainer Video

Tutor: Bilal

Summary

Sequences

In a nutshell

​Sequences can be described in different ways. The term-to-term rule indicates the pattern e.g. +2+2+2​ each time, or ×5\times 5×5 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 1st1st1st difference is the same. The 1st1st1st difference is the difference between adjacent terms.​

​1,6,11,16,211, 6, 11, 16, 211,6,11,16,21​

Geometric

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

​3,6,12,24,483, 6, 12, 24, 483,6,12,24,48​

Periodic

There is a repeated pattern or section.

​1,6,3,1,6,3,11, 6, 3, 1, 6, 3, 11,6,3,1,6,3,1​​

Fibonacci

Each term is the sum of the two previous terms.

​1,1,2,3,5,8,131, 1, 2, 3, 5, 8, 131,1,2,3,5,8,13​​

Quadratic

n2n^2n2  is in the nth term formula. The 2nd2nd2nd​ differences are the same.

​2,5,10,17,262, 5, 10, 17, 262,5,10,17,26



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=4=4, Rule +3+3+3 each time.


​4,7,10,13,16‾\underline{4, 7, 10, 13, 16}4,7,10,13,16​


Example 2

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

First term =−2=-2=−2, Rule ×−2\times -2×−2 each time.


​−2,4,−8,16,−32‾\underline{-2, 4, -8, 16, -32}−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

​1,6,11,16,211, 6, 11, 16, 211,6,11,16,21​​

​+5+5+5 each time​

​3,6,12,24,483, 6, 12, 24, 483,6,12,24,48​​
​×2\times 2×2 each time​

​5,−5,5,−5,55, -5, 5, -5, 55,−5,5,−5,5​​

​×(−1)\times (-1)×(−1) each time​




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Linear number sequences

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Number patterns and sequences

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FAQs - Frequently Asked Questions

What are the different types of sequences?

A number sequence is a set of numbers that follow a pattern. There are different types that can be categorised as arithmetic sequences, geometric sequences, quadratic sequences, fibonacci sequences and periodic sequences.

What is the term-to-term rule in sequences?

The term-to-term rule is a way of defining a sequence. It shows how to calculate the next term from the current term. For example, add 5 each time, or multiply by 3 each time.

What is a sequence?

A sequence is a set of numbers that follow a particular pattern or rule. The order of numbers matters, and the rule can given by a term-to-term rule, or a nth term formula.

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