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Exam questions · Maths · Algebra

Sequences and the nth Term

  • 7 exam questions
  • 19 marks
  • 10 quick checks
  1. 1 Write down [2 marks]

    Here are the first five terms of a sequence. \(7\) \(11\) \(15\) \(19\) \(23\) (a) Write down the next term. [1 mark] (b) Describe the term-to-term rule. [1 mark]

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    Model answer

    (a) The sequence goes up by 4, so the next term is \(23 + 4 = 27\). (b) Add 4 each time.

    Mark scheme

    • (a) 27 — B1
    • (b) Add 4 (to the previous term) — B1
  2. 2 Write down [2 marks]

    Write down an expression for the nth term of the sequence \(6, 9, 12, 15, \ldots\)

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    Model answer

    The difference is 3, so the nth term begins \(3n\). \(3n\) is \(3, 6, 9, 12\), and each term is 3 more, so the nth term is \(3n + 3\).

    Mark scheme

    • \(3n\) seen — M1
    • \(3n + 3\) — A1
  3. 3 Find [2 marks]

    Find the nth term of the sequence \(20, 17, 14, 11, \ldots\)

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    Model answer

    The sequence goes down by 3, so it begins \(-3n\). \(-3n\) is \(-3, -6, -9, -12\), and each term is 23 more. So the nth term is \(-3n + 23\).

    Mark scheme

    • \(-3n\) seen — M1
    • \(-3n + 23\) or \(23 - 3n\) — A1
  4. 4 Work out [3 marks]

    The nth term of a sequence is \(5n - 2\). (a) Work out the 20th term. [1 mark] (b) Is 200 a term in the sequence? You must show how you decide. [2 marks]

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    Model answer

    (a) \(5 \times 20 - 2 = 98\). (b) \(5n - 2 = 200\) gives \(5n = 202\), so \(n = 40.4\). The position must be a whole number, so 200 is not a term.

    Mark scheme

    • (a) 98 — B1
    • (b) \(5n - 2 = 200\) or \(n = 40.4\) — M1
    • (b) No, with a correct reason — Q1
  5. 5 Work out [5 marks]

    Here are the first three patterns in a sequence. Each pattern is made from matchsticks. (a) How many matchsticks are needed for Pattern 5? [1 mark] (b) Find an expression, in terms of \(n\), for the number of matchsticks in Pattern \(n\). [2 marks] (c) Is there a pattern that uses exactly 100 matchsticks? You must show how you decide. [2 marks]

    The first three patterns in a sequence: one square, two squares and three squares in a row, made from matchsticks.
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    Model answer

    (a) The numbers are 4, 7, 10, 13, 16, so Pattern 5 uses 16. (b) The nth term is \(3n + 1\). (c) \(3n + 1 = 100\) gives \(3n = 99\), so \(n = 33\). Yes, Pattern 33 uses exactly 100 matchsticks.

    Mark scheme

    • (a) 16 — B1
    • (b) \(3n\) seen — M1
    • (b) \(3n + 1\) — A1
    • (c) \(3n + 1 = 100\) or \(n = 33\) — M1
    • (c) Yes, Pattern 33 — Q1
  6. 6 Find [3 marks]

    The first five terms of a quadratic sequence are \(2, 9, 20, 35, 54\). Find an expression for the nth term.

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    Model answer

    First differences: 7, 11, 15, 19. Second differences: 4, 4, 4. So the nth term starts \(2n^2\), which is 2, 8, 18, 32, 50. Subtracting from the terms leaves 0, 1, 2, 3, 4, which is \(n - 1\). The nth term is \(2n^2 + n - 1\).

    Mark scheme

    • Second difference 4 found, or \(2n^2\) seen — M1
    • Subtracts \(2n^2\) to leave 0, 1, 2, 3, 4 or \(n - 1\) — M1
    • \(2n^2 + n - 1\) — A1
  7. 7 Work out [2 marks]

    The first two terms of a Fibonacci-type sequence are 3 and 5. Work out the 6th term.

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    Model answer

    Each term is the sum of the two before it: \(3, 5, 8, 13, 21, 34\). The 6th term is 34.

    Mark scheme

    • A correct continuation, such as 8 and 13 — M1
    • 34 — A1

Quick check

  1. 1

    The nth term of a sequence is \(3n + 2\). What is the first term that is greater than 100?

    1. A104
    2. B98
    3. C101
    4. D100
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    C: 101

    \(3n + 2 > 100\) gives \(n > 32.67\ldots\), so \(n = 33\) and the term is \(3 \times 33 + 2 = 101\).

  2. 2

    What is the common ratio of the geometric sequence 5, 10, 20, 40, ...?

    1. A10
    2. B2
    3. C5
    4. D15
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    B: 2

    Each term is the previous term multiplied by 2.

  3. 3

    What is the next term in the sequence 2, 5, 8, 11, ...?

    1. A15
    2. B17
    3. C13
    4. D14
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    D: 14

    The sequence goes up by 3 each time, so the next term is \(11 + 3 = 14\).

  4. 4

    What is the nth term of the sequence 4, 7, 10, 13, ...?

    1. A\(4n + 3\)
    2. B\(3n + 4\)
    3. C\(3n + 1\)
    4. D\(n + 3\)
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    C: \(3n + 1\)

    The difference is 3, so it begins \(3n\). The sequence is 1 more than \(3n\), so the nth term is \(3n + 1\).

  5. 5

    Which sequence has nth term \(2n - 3\)?

    1. A\(-1, 1, 3, 5\)
    2. B\(2, 4, 6, 8\)
    3. C\(5, 7, 9, 11\)
    4. D\(-3, -1, 1, 3\)
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    A: \(-1, 1, 3, 5\)

    Substituting \(n = 1, 2, 3, 4\) gives \(-1, 1, 3, 5\).

  6. 6

    What is the nth term of the sequence 20, 17, 14, 11, ...?

    1. A\(-3n + 23\)
    2. B\(-3n + 20\)
    3. C\(23n - 3\)
    4. D\(3n + 17\)
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    A: \(-3n + 23\)

    The sequence goes down by 3, so it begins \(-3n\). Adding 23 gives 20 when \(n = 1\).

  7. 7

    Which of these numbers is a term in the sequence with nth term \(4n + 1\)?

    1. A63
    2. B61
    3. C62
    4. D64
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    B: 61

    \(4n + 1 = 61\) gives \(n = 15\). The other numbers do not give a whole number for \(n\).

  8. 8

    Which of these is a geometric sequence?

    1. A\(3, 6, 9, 12\)
    2. B\(3, 6, 12, 24\)
    3. C\(1, 4, 9, 16\)
    4. D\(1, 1, 2, 3\)
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    B: \(3, 6, 12, 24\)

    Each term is multiplied by 2: \(3 \times 2 = 6\), \(6 \times 2 = 12\), \(12 \times 2 = 24\).

  9. 9

    The first four terms of a Fibonacci-type sequence are 1, 3, 4, 7. What is the 5th term?

    1. A11
    2. B12
    3. C14
    4. D10
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    A: 11

    Each term is the sum of the two before it, so the 5th term is \(4 + 7 = 11\).

  10. 10

    A sequence has a constant second difference of 2. Its nth term begins with which term?

    1. A\(4n^2\)
    2. B\(n^2\)
    3. C\(2n\)
    4. D\(2n^2\)
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    B: \(n^2\)

    The coefficient of \(n^2\) is half the second difference, so it is \(1\), giving \(n^2\).