Find the common ratio of a sequence, write the nth term arn−1, and solve problems with growth, decay and missing terms. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Objectives
Find the common ratio of a sequence.
Write and use the nth term a x r^(n-1).
Find a missing term or ratio from two known terms.
Key points
Test a sequence by dividing each term by the one before it to find the common ratio.
Continue a geometric sequence forwards by multiplying and backwards by dividing.
Write and use the nth term un=arn−1.
Find an unknown ratio or first term from two terms, and link percentage growth and decay to the ratio.
Worked examples
Example 1
Find the common ratio of the sequence 4, 12, 36, 108.
2 marks
Divide the second term by the first: 12 / 4
12 / 4 = 3
Check: 36 / 12 = 3, so r = 3
Answer: 3
Common slips
Slip: Subtract the first term from the second: 12 - 4 A ratio comes from dividing, not subtracting.
Slip: 12 - 4 = 8 Divide the terms.
Slip: Check: 36 - 12 = 24 The same ratio should appear again.
Example 2
A geometric sequence starts 5, 10, 20, 40. Work out the 8th term.
2 marks
The first term is 5 and the ratio is 2
The nth term is 5 x 2^(n-1), so the 8th term is 5 x 2^7
5 x 128 = 640
Answer: 640
Common slips
Slip: The first term is 10 The first term is the first number in the list.
Slip: The 8th term is 5 x 2^8 The power is one less than the position.
Slip: (5 x 2)^7 The power applies to the ratio only.
Example 3
A geometric sequence has second term 12 and fifth term 324, with ratio 3. Find the first term.
3 marks
Going from the first term to the second multiplies by the ratio
So the first term is the second term divided by 3
The first term is 4, because 12 divided by 3 gives 4
Answer: 4
Common slips
Slip: The second term is the first term plus 3 A geometric sequence multiplies.
Slip: So the first term is the second term times 3 Going back divides by r.
Slip: 12 x 3 = 36 Divide to go back one place.
Questions and answers
GCSE Maths geometric sequences questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Find the common ratio of 2, 10, 50, 250.
Show answer
Answer: 5
Divide 10 by 2.
Check with 50 divided by 10.
M1 — Divide 10 by 2.
A1 — Answer 5
Question 2 Easy2 marks
Find the common ratio of 80, 40, 20, 10.
Show answer
Answer: 1/2
Divide 40 by 80.
The ratio is a fraction because the terms shrink.
M1 — Divide 40 by 80.
A1 — Answer 1/2
Question 3 Medium2 marks
A geometric sequence has first term 2 and common ratio 3. Work out the 6th term.
Show answer
Answer: 486
The power is 6 - 1 = 5.
2 x 3^5.
M1 — The power is 6 - 1 = 5.
A1 — Answer 486
Question 4 Medium2 marks
The first term is 7 and the common ratio is 2. Work out the 5th term.
Show answer
Answer: 112
2^4 = 16.
7 x 16.
M1 — 2^4 = 16.
A1 — Answer 112
Question 5 Hard2 marks
A geometric sequence has second term 6 and fourth term 54, and a positive ratio. Find the common ratio.
Show answer
Answer: 3
Two steps: r squared = 54 / 6 = 9.
r = 3.
M1 — Two steps: r squared = 54 / 6 = 9.
A1 — Answer 3
Question 6 Hard2 marks
The 3rd term of a geometric sequence is 20 and the common ratio is 2. Find the first term.
Show answer
Answer: 5
Go back two places: divide by 2 twice.
20 / 4.
M1 — Go back two places: divide by 2 twice.
A1 — Answer 5
Question 7 Exam-style4 marks
The first three terms of a geometric sequence are 6, x and 54, where x is positive. Work out x and the 6th term of the sequence.
Show answer
Answer: 18 and 1458
x over 6 equals 54 over x, so x squared is 324.
x = 18 and r = 3.
The 6th term is 6 x 3^5.
6 x 243 = 1458.
M1 — Sets x/6 = 54/x or x^2 = 6 x 54
A1 — Finds that x is 18
M1 — Uses first term times ratio to the power 5
A1 — The 6th term is 1458
Common mistakes
Most lost marks on this topic are one slip.
Using the power n instead of n - 1 in the nth term
Put n = 1 into the formula to check it gives the first term
Multiplying a and r together before applying the power
Apply the power to r only, then multiply by a
Finding the ratio by subtracting terms
Divide a term by the one before it
Spot the slip
A student says 3, 7, 11, 15 is geometric because the numbers keep growing. What is wrong?
A geometric sequence multiplies by the same number, and this one adds 4
Growing sequences are always geometric
The terms should double
Show the slip
A geometric sequence multiplies by the same number, and this one adds 4 Adding the same number each time makes an arithmetic sequence. Geometric means multiplying by a common ratio.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.