# Geometric Sequences

Let Beyond Revision be the constant multiplier in your learning journey with this walkthrough on geometric sequences.

Whether you’re running through the topic for the first time at KS3 level or revising the fundamentals for your GCSE revision, this guide should work a treat.

### What are Geometric Sequences?

In a geometric sequence, the terms (each number in the sequence) increase or decrease by a constant multiplier. This means you find the next term by multiplying the term before by a fixed value.

For example, the following geometric sequence has a multiplier of 2:

### Example 1:

Let’s look at the following geometric sequence:

• 9, 27, 81, 243, 729…

We can choose any two adjacent terms and divide the larger number by the smaller one.

We could choose 243 and 729:

• 729 ÷ 243 = 3

Similarly, we could choose 27 and 81 and we would get the same multiplier:

• 81 ÷ 27 = 3

The multiplier is 3.

Multiple your learning potential with our Walkthrough Worksheet on geometric sequences…

### Example 2:

Write down the next term in the following sequence:

• 6, 30, 150, 750, 3750, …….

Sometimes, you might be asked to write down the next term(s) in a sequence.

We would begin by finding the multiplier:

• 30 ÷ 6 = 5

Now, multiply the last known term in the sequence (3750) by the multiplier (5):

• 3750 × 5 = 18 750

### Example 3:

Write down the next term in the following sequence:

• 15 000, 3000, 600, 120, …….

This time the sequence is descending. We now have two options:

Option 1

Divide a larger term by its adjacent smaller term:

• 15 000 ÷ 3000 = 5

This means to find the next term in the sequence, we divide the last known term by 5:

• 120 ÷ 5 = 24

Option 2

Divide a smaller term by its adjacent larger term:

• 120 ÷ 600 = $\frac{1}{5}$ or 0.2

This means to find the next term in the sequence, we multiply the last known term by $\frac{1}{5}$ or 0.2:

• 120 × $\frac{1}{5}$ = 24

Either way, the next term in the sequence is 24.

### Practice Questions

1. Write down the multiplier for each sequence.
a)
1, 3, 9, 27, 81
b) 1, 6, 36, 216
c) 2, 6, 18, 54
d) 1, 7, 49, 343
e) 20, 30, 45, 67.5
f) 2, 10, 50, 250
g) 14 400, 3600, 900
h) 8, 56, 112, 224

2. Write down the next two terms in each sequence:
a)
4, 12, 36, 108…
b) 8, 16, 32, 64…
c) 15, 45, 135, 405…
d) 96 000, 24 000, 6000, 1500…
e) 56 320, 7040, 880, 110…
f) 6, 15, 37.5, 93.75…
g) 5, 7.5, 11.25, 16.875…
h) 12.5, 2.5, 0.5, 0.1…

Challenge

Taylor purchased 5 books. The first book cost 65p, the second book cost £1.95 and the third book cost £5.85. How much did they pay for all 5 books? (Assume that the price of subsequent books continues to rise using the same multiplier).

1. Write down the multiplier for each sequence.
a)
1, 3, 9, 27, 81 = 3
b) 1, 6, 36, 216 = 6
c) 2, 6, 18, 54 = 3
d) 1, 7, 49, 343 = 7
e) 20, 30, 45, 67.5 = 1.5
f) 2, 10, 50, 250 = 5
g) 14 400, 3600, 900 = x 0.25 or ÷ 4
h) 8, 56, 112, 224 = 2

2. Write down the next two terms in each sequence:
a)
4, 12, 36, 108… = 324,972
b) 8, 16, 32, 64… = 128,256
c) 15, 45, 135, 405… = 1215,3645
d) 96 000, 24 000, 6000, 1500… = 375, 93.75
e) 56 320, 7040, 880, 110… = 13.75, 1.71875
f) 6, 15, 37.5, 93.75… = 234.375, 585.9375
g) 5, 7.5, 11.25, 16.875… = 25.3125, 37.96875
h) 12.5, 2.5, 0.5, 0.1… = 0.02, 0.004

Challenge

Taylor purchased 5 books. The first book cost 65p, the second book cost £1.95 and the third book cost £5.85. How much did they pay for all 5 books? (Assume that the price of subsequent books continues to rise using the same multiplier).

Solution =

Book 1: £0.65

Book 2: £1.95

Book 3: £5.85

Constant multiplier

Book 4: £17.55

Book 5: £52.65

0.65 + 1.95 + 5.85 + 17.55 + 52.65 = £78.65

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