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Exam questions · Maths · Functions, Sequences and Rates of Change

Geometric and Special Sequences

  • 6 exam questions
  • 16 marks
  • 9 quick checks
  1. 1 Calculate [3 marks]

    Here are the first four terms of a geometric sequence: \(800, 400, 200, 100\) (a) Write down the common ratio. [1 mark] (b) Write down the next two terms. [2 marks]

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    Model answer

    (a) \(400 \div 800 = \dfrac{1}{2}\). (b) \(100 \div 2 = 50\) and \(50 \div 2 = 25\).

    Mark scheme

    • (a) \(\dfrac{1}{2}\) or \(0.5\) — B1
    • (b) \(50\) — B1
    • (b) \(25\) — B1
  2. 2 Calculate [2 marks]

    The first two terms of a sequence are 1 and 4. Each term after that is the sum of the two terms before it. Calculate the 6th term. [2 marks]

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    Model answer

    The terms are \(1, 4, 5, 9, 14, 23\), so the 6th term is 23.

    Mark scheme

    • Continues the sequence, \(5, 9, 14\) — M1
    • \(23\) — A1
  3. 3 Calculate [2 marks]

    The first term of a geometric sequence is 2 and the common ratio is \(-3\). Calculate the 4th term. [2 marks]

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    Model answer

    \(2, -6, 18, -54\), so the 4th term is \(-54\).

    Mark scheme

    • \(2 \times (-3)^3\) or \(2, -6, 18\) — M1
    • \(-54\) — A1
  4. 4 Calculate [3 marks]

    The 2nd term of a geometric sequence is 20 and the 5th term is 2.5. Calculate the first term. [3 marks]

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    Model answer

    There are 3 steps from the 2nd to the 5th term, so \(r^3 = \dfrac{2.5}{20} = \dfrac{1}{8}\) and \(r = \dfrac{1}{2}\). The first term is \(20 \div \dfrac{1}{2} = 40\).

    Mark scheme

    • \(r^3 = \dfrac{2.5}{20} = \dfrac{1}{8}\) — M1
    • \(r = \dfrac{1}{2}\) — A1
    • \(40\) — A1
  5. 5 Calculate [3 marks]

    The first three terms of a geometric sequence are \(5, 5\sqrt{5}, 25\). Calculate the 4th term. [3 marks]

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    Model answer

    The common ratio is \(\sqrt{5}\). The 4th term is \(25 \times \sqrt{5} = 25\sqrt{5}\).

    Mark scheme

    • Common ratio \(\sqrt{5}\) — B1
    • \(25 \times \sqrt{5}\) — M1
    • \(25\sqrt{5}\) — A1
  6. 6 Calculate [3 marks]

    \(3, x, 48\) are three consecutive terms of a geometric sequence. All the terms are positive. Calculate the value of \(x\). [3 marks]

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    Model answer

    \(\dfrac{x}{3} = \dfrac{48}{x}\), so \(x^2 = 144\) and \(x = 12\).

    Mark scheme

    • \(\dfrac{x}{3} = \dfrac{48}{x}\) — M1
    • \(x^2 = 144\) — M1
    • \(12\) — A1

Quick check

  1. 1

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

    1. A9
    2. B3
    3. C4
    4. D36
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    C: 4

    \(12 \div 3 = 4\).

  2. 2

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

    1. A108
    2. B162
    3. C72
    4. D216
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    B: 162

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

  3. 3

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

    1. A21
    2. B18
    3. C26
    4. D16
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    A: 21

    \(8 + 13 = 21\).

  4. 4

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

    1. A2
    2. B\(-2\)
    3. C\(\dfrac{1}{4}\)
    4. D\(\dfrac{1}{2}\)
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    D: \(\dfrac{1}{2}\)

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

  5. 5

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

    1. A54
    2. B486
    3. C162
    4. D90
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    C: 162

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

  6. 6

    Which of these sequences is geometric?

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

    Each term is multiplied by 3.

  7. 7

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

    1. A2
    2. B3
    3. C8
    4. D4
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    A: 2

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

  8. 8

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

    1. A3
    2. B\(2\sqrt{3}\)
    3. C\(\dfrac{1}{\sqrt{3}}\)
    4. D\(\sqrt{3}\)
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    D: \(\sqrt{3}\)

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

  9. 9

    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\(3 \times 2^{n-1}\)
    4. D\(2 \times 3^{n-1}\)
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    C: \(3 \times 2^{n-1}\)

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