# GCSE Maths Understanding Sequences

**Introduction**

As a part of your GCSE Maths Understanding Sequences content you will be looking at numbers and the patterns amongst different types of numbers. This is known as sequences and it is generally the case that you add a certain number to get to the next number, or you subtract a certain number or you multiply or divide.

Essentially you are trying to find what number comes next. Or even what the next four terms of a particular sequence will be.

In this articles we are going to discuss how sequences develop and show you how to identify the nth term of any given linear sequence.

## GCSE Maths Understanding Sequences - Overview

A **sequence** is a set of numbers, but there is a **specific rule** which is used to generate those numbers.

The rule could be adding by a fixed amount of multiplication of a fixed amount or something else.

If you consider the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, then the rule here is to +1 to get to the next number. Each number in a sequence is called a **term**.

**Spotting Patterns**

Much of the work involved in sequences does involve spotting a relationship or a pattern within that sequence.

Consider the sequence 1, 3, 6, 10, 15

What is the pattern here?

Well if we look at the **difference **between the numbers we have **+2, +3, +4 **and** +5. **The pattern here is that the difference is increasing by 1 every time.

Based on this if we want to find the next term then we need to +6 to 15 to give 21.

If we want the following term then we need to +7 to 21 to give 28.

## Example

**Given the sequence 1, 1, 2, 3, 5, 8, 13**

**Find the next two terms. **

**Solution:**

**First we need to determine what the pattern is. If we take the difference between numbers we have 0, 1, 1, 2, 3, 5.**

**So we cannot say that the difference is increasing by a fixed amount each time.**

**But if you add the first two terms you get the third term:**

**1 + 1 = 2**

**And if we continue like this:**

**1 + 2 = 3**

**2 + 3 = 5**

**3 + 5 = 8**

**5 + 8 = 13**

**So to find the next term we must add 13 with 8 to give ****21**

**And to determine the second term we must add 21 with 13 to give ****34**

**Determining a sequence if given a general rule**

It is quite common for a sequence to be defined by a general rule and from that we can determine the terms of its sequence. The rule is usually expressed in the form of “n”, where n is the position of the term in the sequence.

**Example:**

**A sequence is defined by **3n+2**. Write down the first three terms. **

**Solution:**

**In questions such as this, **n**, is defined for all positive numbers. So **n=1, 2, 3, 4, 5, 6 etc

**To find the first three terms of the sequence we substitute **n=1, 2, 3** into the rule**

n=1**: 3(1) + 2 = 5**

n=2**: 3(2) + 2 = 8 **

n=3**: 3(3) + 2 = 11**

**So 5, 8, 11 are the first three terms of the sequence. **

**Example:**

**A sequence is defined by **4n-3**. Find the 4****th**** term. **

**Solution:**

**Because we are only interested in the 4****th**** term this means that **n=4**. So we just need to substitute this value into the defined rule. **

**4****th**** term = 4(4) – 3 = 16 – 3 = 13. **

**Finding the **n**th term of a sequence**

The nth term of a sequence is always of the form An+b.

**Example:**

**Find the **n**th term of the sequence; 5,7,9,11,13**

**Solution:**

**First we need to determine how the sequence is increasing and hopefully by spotting you can see that this is increasing by 2 each time. **

**Because of this, we say that each term is a multiple of **2n**. **

**Next we need to look at the first term of the sequence which is 5 and we need to subtract the difference between each term of the sequence, i.e. 5 – 2 = 3**

**So the **n**th term of the sequence is **2n+3

**Check: **

n=1**: 2(1) + 3 = 5**

n=2**: 2(2) + 3 = 7**

n=3**: 2(3) + 3 = 9**

**This check shows that we have matched the first three terms of our sequence**

The good thing about sequences is that you can check your answer to see if you are correct, which is good to know. As long as you are careful with your arithmetic and that you show all your work in terms of how you get to a particular answer then you ought to find GCSE Maths Sequences quite an easy topic and one where you can be confident to quickly pick up full marks so make sure you are getting plenty of question practice done in this area.