You may use a calculator in this activity.

This activity is about recognising and continuing **geometric sequences**.

Imagine the situation below.

Matt suggests to his parents: *"Let's change how much pocket money I get! How about you give me 1p the first week, 2p the second week, 4p the third week, 8p in week 4, 16p in week 5 and so on...?"*

Starting at 1, we get the next number by **multiplying by 2** or **doubling the previous number**.

The sequence of Matt's weekly pocket money in pence:

1, 2, 4, 8, 16 ...

These numbers are called the **powers of 2**.

Week 1... 2^{0} = 1

Week 2... 2^{1} = 2

Week 3... 2^{2} = 4

Week 4... 2^{3} = 8

Week 5... 2^{4} = 16

The** 'powers of 2' **are an example of a **geometric **sequence, where each term after the first is found by multiplying the previous term by a fixed number, called the common ratio.

The common ratio here is 2.

Let's look at a different geometric sequence!

## 1, 3, 9, 27, ....

What is the common ratio and what is the next term?

**Method**

To find the common ratio (which means, what does the sequence go up in), we can divide any term by its previous one.

For example 3 ÷ 1 = 3

or we could use 9 ÷ 3 = 3

or the 4th term, 27 ÷ 9 = 3

All of them give us a **common ratio of 3**

That means to get the next term we multiply the previous term by 3 and the rule increases by powers of 3:

4th term is 27

5th term = 27 x 3 = 81

1st Term ... 3^{0} = 1

2nd Term ... 3^{1} = 3

3rd Term ... 3^{2} = 9

4th Term ... 3^{3} = 27

5th Term... 3^{4} = 81

Notice that **the power is one less than the term**, ie, the 5th term is the power of 4, or the 3rd term is the power of 2.

Let's do some practise!