[discia] GCSE Maths › Algebra › Geometric Sequences

Geometric Sequences

discia.org 10 questions · 40 marks
Name Date
  1. Either paper 3 marks

    Here are the first four terms of a geometric sequence.

    \(4,\ 16,\ 64,\ 256\)

    (a)

    Write down the common ratio of the sequence.

    (b)

    Work out the next two terms of the sequence.

    Show answer
    (a) \(4\)
    (b) \(1,024,\ 4,096\)
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  2. Either paper 4 marks
    \(A\) is an arithmetic sequence. \(G\) is a geometric sequence. Here are the first four terms of each sequence.
    Term number1st2nd3rd4th
    \(A\)53777491121
    \(G\)24816
    Work out the smallest value of \(n\) for which the \(n\)th term of \(G\) is more than 3716 greater than the \(n\)th term of \(A\).
    Show answer
    \(13\)
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  3. Either paper 4 marks

    Here are the first four terms of a geometric sequence.

    \(9,\ 27,\ 81,\ 243\)

    (a)

    Find an expression, in terms of \(n\), for the \(n\)th term of this sequence.

    (b)

    Work out the 7th term of the sequence.

    Show answer
    (a) \(9 \times 3^{n-1}\)
    (b) \(6,561\)
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  4. Either paper 3 marks

    Here are the first four terms of a geometric sequence.

    \(896,\ 224,\ 56,\ 14\)

    (a)

    Write down the common ratio of the sequence.

    (b)

    Find the fifth term of the sequence.

    Give your answer as a fraction.

    Show answer
    (a) \(\frac{1}{4}\)
    (b) \(\frac{7}{2}\)
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  5. Either paper 5 marks

    The number of ants in a nest is multiplied by 2 every week.

    At the start there are 32 ants.

    (a)

    Work out how many ants there will be after 2 weeks.

    (b)

    Work out after how many weeks the number of ants first becomes greater than 8,200.

    Show answer
    (a) \(128\)
    (b) \(9\ \text{weeks}\)
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  6. Either paper 6 marks

    The terms of a geometric progression are all positive.

    Here are the second and fourth terms, with the first and third terms missing.

    \(\ldots,\ 16,\ \ldots,\ 4\)

    (a)

    Work out the first term.

    (b)

    Work out the third term.

    (c)

    The first two terms of an arithmetic progression are shown.

    \(p,\ 4p,\ \ldots\)

    The sum of the first three terms is \(144\).

    Work out the value of \(p\).

    Show answer
    (a) \(32\)
    (b) \(8\)
    (c) \(12\)
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  7. Either paper 4 marks

    Here are the first four terms of a geometric sequence.

    \(8,\ -24,\ 72,\ -216\)

    (a)

    State the common ratio of the sequence.

    (b)

    Calculate the seventh term of the sequence.

    (c)

    Is the 94th term of this sequence positive or negative?

    Explain your answer.

    Show answer
    (a) \(-3\)
    (b) \(5,832\)
    (c) It is negative, because 94 is even and the terms alternate: every odd-numbered term is positive and every even-numbered term is negative.
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  8. Non-calculator 4 marks
    (a)

    The first three terms of a geometric progression are

    \(3\sqrt{ 3 }\), \(9\), \(9\sqrt{ 3 }\)

    Work out the next term.

    (b)

    The \(n\)th term of a different sequence is \(\left(2 + \sqrt{ 7 }\right)^n\)

    Show that the third term is \(50 + 19\sqrt{ 7 }\)

    Show answer
    (a) \(27\)
    (b) Squaring gives \(11 + 4\sqrt{ 7 }\), and multiplying by \(2 + \sqrt{ 7 }\) gives \(50 + 19\sqrt{ 7 }\), which is the printed third term.
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  9. Either paper 3 marks

    Here are the first four terms of a geometric sequence.

    \(10,\ 40,\ 160,\ 640\)

    (a)

    Write down the common ratio of the sequence.

    (b)

    Work out the next two terms of the sequence.

    Show answer
    (a) \(4\)
    (b) \(2,560,\ 10,240\)
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  10. Either paper 4 marks
    \(A\) is an arithmetic sequence. \(G\) is a geometric sequence. Here are the first four terms of each sequence.
    Term number1st2nd3rd4th
    \(A\)133546951036
    \(G\)1248
    Work out the smallest value of \(n\) for which the \(n\)th term of \(G\) is greater than the \(n\)th term of \(A\).
    Show answer
    \(14\)
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discia.org Geometric Sequences · sheet 1
Answers Geometric Sequences · sheet 1
  1. (a) \(4\)
    (b) \(1,024,\ 4,096\)
  2. \(13\)
  3. (a) \(9 \times 3^{n-1}\)
    (b) \(6,561\)
  4. (a) \(\frac{1}{4}\)
    (b) \(\frac{7}{2}\)
  5. (a) \(128\)
    (b) \(9\ \text{weeks}\)
  6. (a) \(32\)
    (b) \(8\)
    (c) \(12\)
  7. (a) \(-3\)
    (b) \(5,832\)
    (c) It is negative, because 94 is even and the terms alternate: every odd-numbered term is positive and every even-numbered term is negative.
  8. (a) \(27\)
    (b) Squaring gives \(11 + 4\sqrt{ 7 }\), and multiplying by \(2 + \sqrt{ 7 }\) gives \(50 + 19\sqrt{ 7 }\), which is the printed third term.
  9. (a) \(4\)
    (b) \(2,560,\ 10,240\)
  10. \(14\)
discia.org Geometric Sequences · sheet 1