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Maths · Functions, Sequences and Rates of Change

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Geometric and Special Sequences

Recognising geometric and Fibonacci-type sequences, and finding the common ratio from two terms.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Recognise a geometric sequence and find its common ratio.
  2. 2Find terms of a geometric sequence, and use the term-to-term rule.
  3. 3Recognise Fibonacci-type, square, cube and triangular number sequences.
  4. 4Find the common ratio and first term from two given terms (Higher tier), including a surd ratio.

Sequences that multiply

In an arithmetic sequence, each term is found by adding the same number. In a geometric sequence, each term is found by multiplying by the same number, which is called the common ratio. Geometric sequences grow or shrink very quickly. Basic geometric sequences with a whole-number or fractional ratio can be set on both tiers, but finding a ratio from two terms, and using a surd as the ratio, are Higher tier skills. Other special sequences are also worth knowing, such as Fibonacci-type sequences.

The \(n\)th term of a geometric sequence

To get from the first term to the \(n\)th term you multiply by the ratio \(n - 1\) times.

  • Formula

    \(n\)th term \(= a \times r^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.

  • Example

    For \(2, 6, 18, \ldots\), the \(n\)th term is \(2 \times 3^{n-1}\).

  • Check

    The 4th term is \(2 \times 3^3 = 54\).

  • Not the same as arithmetic

    An arithmetic sequence has an \(n\)th term of the form \(an + b\).

Finding terms

A geometric sequence starts \(5, 10, 20, 40, \ldots\). Work out the 8th term.

Show the solutionHide the solution
  1. 1 Common ratio \(r = \dfrac{10}{5} = 2\).
  2. 2 Formula \(n\)th term \(= 5 \times 2^{n-1}\).
  3. 3 For the 8th term \(5 \times 2^7 = 5 \times 128\).
  4. 4 Answer 640.

Answer640

Finding the ratio and the first term (Higher tier)

Two terms that are not next to each other are enough to find the ratio.

  • Compare the terms

    If the 2nd term is 6 and the 5th term is 48, there are 3 steps between them, so \(r^3 = \dfrac{48}{6} = 8\).

  • The ratio

    \(r = \sqrt[3]{8} = 2\).

  • The first term

    Divide back: \(6 \div 2 = 3\).

  • A surd ratio

    A ratio can be a surd, such as \(2, 2\sqrt{3}, 6, 6\sqrt{3}\) with \(r = \sqrt{3}\).

A surd ratio (Higher tier)

The first three terms of a geometric sequence are \(4, 4\sqrt{2}, 8\). Find the 6th term.

Show the solutionHide the solution
  1. 1 Ratio \(r = \dfrac{4\sqrt{2}}{4} = \sqrt{2}\).
  2. 2 Next terms \(8\sqrt{2}\), then \(8\sqrt{2} \times \sqrt{2} = 16\).
  3. 3 The 6th term \(16 \times \sqrt{2} = 16\sqrt{2}\).
  4. 4 Check Every two terms multiply by \(\sqrt{2} \times \sqrt{2} = 2\).

Answer\(16\sqrt{2}\)

Other special sequences

Some sequences have names, and you should recognise them.

  • Fibonacci-type

    Each term is the sum of the two terms before it, such as \(1, 1, 2, 3, 5, 8, 13, \ldots\).

  • Square numbers

    \(1, 4, 9, 16, 25, \ldots\), with an \(n\)th term of \(n^2\).

  • Cube numbers

    \(1, 8, 27, 64, \ldots\), with an \(n\)th term of \(n^3\).

  • Triangular numbers

    \(1, 3, 6, 10, 15, \ldots\), where each term adds one more than the last increase.

Test yourself

  1. 1

    What is the common ratio of \(3, 12, 48, 192\)?

    Show answerHide answer

    4.

  2. 2

    What is the \(n\)th term of a geometric sequence?

    Show answerHide answer

    \(a \times r^{n-1}\).

  3. 3

    How do you find the ratio from two terms with \(k\) steps between them?

    Show answerHide answer

    Divide the terms, then take the \(k\)th root.

  4. 4

    What is a Fibonacci-type sequence?

    Show answerHide answer

    Each term is the sum of the previous two.

  5. 5

    What are the first four triangular numbers?

    Show answerHide answer

    1, 3, 6, 10.

Exam technique: sequences

Name the sequence first, and then use the right method.

  • Test the type

    Check differences for arithmetic and ratios for geometric.

  • Show the ratio

    Write \(\dfrac{6}{2} = 3\) so that the method mark is clear.

  • Check the next term

    Test your rule on the terms that you have.

  • Unequal gaps

    If the terms are not next to each other, count the steps between them.

Summary and exam focus

  • In a geometric sequence you multiply by a common ratio each time.
  • The \(n\)th term is \(a \times r^{n-1}\).
  • Find a ratio from two terms with \(r^k\) as the quotient, then take the root (Higher tier).
  • Know Fibonacci-type, square, cube and triangular sequences.

Exam focus

The 2nd term of a geometric sequence is 6 and the 5th term is 48. Work out the first term. (3 marks) (3 marks)

There are 3 steps from the 2nd to the 5th term, so \(r^3 = \dfrac{48}{6} = 8\) and \(r = 2\). The first term is \(6 \div 2 = 3\).

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Geometric sequence
A sequence with a common ratio between terms.
Common ratio
The number each term is multiplied by.
Term-to-term rule
A rule that gives each term from the one before it.
Fibonacci-type sequence
A sequence where each term is the sum of the two before.
Triangular numbers
The sequence 1, 3, 6, 10, 15, and so on.
Square numbers
The numbers 1, 4, 9, 16, and so on.
Cube numbers
The numbers 1, 8, 27, 64, and so on.
Arithmetic sequence
A sequence with a common difference between terms.
Surd
A root that cannot be written as a whole number or fraction.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 3 marks Core

Here are the first four terms of a geometric sequence: \(3, 12, 48, 192\) (a) Write down the common ratio. [1 mark] (b) Work out the next two terms. [2 marks]

Mark scheme — 3 marks available

  • (a) \(4\) — B1
  • (b) \(768\) — B1
  • (b) \(3072\) — B1

Model answer

(a) \(12 \div 3 = 4\). (b) \(192 \times 4 = 768\) and \(768 \times 4 = 3072\).

2. Exam question Work out 2 marks Core

The first two terms of a sequence are 2 and 7. Each term after that is the sum of the two terms before it. Work out the 6th term. [2 marks]

Mark scheme — 2 marks available

  • Continues the sequence, \(9, 16, 25\) — M1
  • \(41\) — A1

Model answer

The terms are \(2, 7, 9, 16, 25, 41\), so the 6th term is 41.

3. Exam question Work out 2 marks Core

The first term of a geometric sequence is 81 and the common ratio is \(\dfrac{1}{3}\). Work out the 4th term. [2 marks]

Mark scheme — 2 marks available

  • \(81 \times \left(\dfrac{1}{3}\right)^3\) or \(81, 27, 9\) — M1
  • \(3\) — A1

Model answer

\(81, 27, 9, 3\), so the 4th term is 3.

4. Exam question Work out 3 marks Core

The 2nd term of a geometric sequence is 12 and the 5th term is 324. Work out the first term. [3 marks]

Mark scheme — 3 marks available

  • \(r^3 = \dfrac{324}{12} = 27\) — M1
  • \(r = 3\) — A1
  • \(4\) — A1

Model answer

There are 3 steps from the 2nd to the 5th term, so \(r^3 = \dfrac{324}{12} = 27\) and \(r = 3\). The first term is \(12 \div 3 = 4\).

5. Exam question Work out 3 marks Stretch

The first four terms of a geometric sequence are \(4, 4\sqrt{2}, 8, 8\sqrt{2}\). Work out the 6th term. [3 marks]

Mark scheme — 3 marks available

  • Common ratio \(\sqrt{2}\) — B1
  • \(8\sqrt{2} \times \sqrt{2} = 16\) for the 5th term — M1
  • \(16\sqrt{2}\) — A1

Model answer

The common ratio is \(\sqrt{2}\). The 5th term is \(8\sqrt{2} \times \sqrt{2} = 16\), and the 6th term is \(16\sqrt{2}\).

6. Exam question Work out 3 marks Stretch

\(2, x, 18\) are three consecutive terms of a geometric sequence. All the terms are positive. Work out the value of \(x\). [3 marks]

Mark scheme — 3 marks available

  • \(\dfrac{x}{2} = \dfrac{18}{x}\) — M1
  • \(x^2 = 36\) — M1
  • \(6\) — A1

Model answer

\(\dfrac{x}{2} = \dfrac{18}{x}\), so \(x^2 = 36\) and \(x = 6\).

7. Multiple choice 1 mark Easier

What is the common ratio of \(3, 12, 48, 192\)?

  1. A 9
  2. B 3
  3. C 4 Correct
  4. D 36

Why: \(12 \div 3 = 4\).

8. Multiple choice 1 mark Easier

What is the next term of \(2, 6, 18, 54\)?

  1. A 108
  2. B 162 Correct
  3. C 72
  4. D 216

Why: Multiply by 3: \(54 \times 3 = 162\).

9. Multiple choice 1 mark Easier

What is the next term in the Fibonacci-type sequence \(3, 5, 8, 13\)?

  1. A 21 Correct
  2. B 18
  3. C 26
  4. D 16

Why: \(8 + 13 = 21\).

10. Multiple choice 1 mark Core

What is the common ratio of \(80, 40, 20, 10\)?

  1. A 2
  2. B \(-2\)
  3. C \(\dfrac{1}{4}\)
  4. D \(\dfrac{1}{2}\) Correct

Why: \(40 \div 80 = \dfrac{1}{2}\).

11. Multiple choice 1 mark Core

What is the 5th term of the geometric sequence \(2, 6, 18, \ldots\)?

  1. A 54
  2. B 486
  3. C 162 Correct
  4. D 90

Why: \(2 \times 3^4 = 162\).

12. Multiple choice 1 mark Core

Which of these sequences is geometric?

  1. A \(1, 3, 5, 7\)
  2. B \(1, 3, 9, 27\) Correct
  3. C \(1, 4, 9, 16\)
  4. D \(1, 1, 2, 3\)

Why: Each term is multiplied by 3.

13. Multiple choice 1 mark Stretch

The 2nd term of a geometric sequence is 6 and the 5th term is 48. What is the common ratio?

  1. A 2 Correct
  2. B 3
  3. C 8
  4. D 4

Why: \(r^3 = \dfrac{48}{6} = 8\), so \(r = 2\).

14. Multiple choice 1 mark Stretch

What is the common ratio of \(2, 2\sqrt{3}, 6, 6\sqrt{3}\)?

  1. A 3
  2. B \(2\sqrt{3}\)
  3. C \(\dfrac{1}{\sqrt{3}}\)
  4. D \(\sqrt{3}\) Correct

Why: \(\dfrac{2\sqrt{3}}{2} = \sqrt{3}\).

15. Multiple choice 1 mark Stretch

A geometric sequence has first term 3 and common ratio 2. What is the \(n\)th term?

  1. A \(3 \times 2^n\)
  2. B \(3n + 2\)
  3. C \(3 \times 2^{n-1}\) Correct
  4. D \(2 \times 3^{n-1}\)

Why: The \(n\)th term is \(ar^{n-1}\).