Use the Geometric Sequence of the Powers of 2 to Double Your Pocket Money

You may use a calculator in this activity.

 

This activity is about Mathematical Matt, who asks 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...?"

 

Here's the weekly amount in pence:

 

1    2    4    8    16  ...

 

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

These numbers are also called the powers of 2.

 

Week 1... 2⁰ = 1

Week 2... 2¹ = 2

Week 3... 2² = 4

Week 4... 2³ = 8

Week 5... 2⁴ = 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 is 2.

 

The powers of 2 sequence goes on forever - it is infinite and grows quickly!

 

Example 1

How much will Matt get, in pence, in week 7?

 

Answer

He will get 2⁶ = 64p.

 

Example 2

How much will Matt get, in pence, in total by week 4?

 

Answer

He will get 1 + 2 + 4 + 8 = 15p.

 

Example 3

How much will Matt get, in pounds, in week 17?

 

Answer

He will get 2¹⁶ = 65,536p = £655.36.

 

Shall we get started?