GCSE Maths revision · Higher: number and algebra

Geometric sequences

Higher only

The idea

A geometric sequence multiplies by the same ratio $r$ each time. The ratio can be a fraction, negative, or a surd: $\sqrt{2}, 2, 2\sqrt{2}, 4$ has $r = \sqrt{2}$ .

Divide a term by the one before to find $r$ .
Check it is the same between every pair.
Multiply by $r$ to continue; the $n$ th term is $ar^{n-1}$ .

Watch out A geometric sequence has a common ratio, not a common difference.

See it worked

Worked exampleNon-calculator3 marks

Here are the first four terms of a geometric sequence: $\sqrt{2}, \ 2, \ 2\sqrt{2}, \ 4$ .

(a) Write down the common ratio. Give your answer in exact form.

(b) Work out the 8th term.

(a) $\frac{2}{\sqrt{2}} = \sqrt{2}$ : each term is $\sqrt{2}$ times the one before.
(b) The $n$ th term is $(\sqrt{2})^n$ .
$(\sqrt{2})^8 = 2^4 = 16$

Answer (a) $\sqrt{2}$ ; (b) $16$

Your turn

Question 1Calculator allowed2 marks

A geometric sequence has first term 3 and common ratio $\sqrt{2}.$ Find the sixth term in the form $a\sqrt{2}.$

Question 2Non-calculator4 marks

A geometric sequence has second term 8 and fifth term 64. Its common ratio is positive.

(a) Find the common ratio.

(b) Find the first term.

Question 3Non-calculator4 marks

In a sequence, each term after the second is the sum of the two terms before it. The first term is $a$ and the second term is $b$ . The fifth term is 19 and the sixth term is 31.

Find the values of $a$ and $b$ . Give your answer as $(a, b)$ .

In past papers

9 official questions in 9 papers since 2018. The latest:

All GCSE Maths past papers

GCSE Maths Pro

£119 once

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.

Start free

No VAT is charged.