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Sequences and series: OCR MEI A‑Level Maths questions

Every OCR MEI A-Level Maths exam question on P.4 Sequences and series, 2018–2025, with its mark scheme and examiner’s report.

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560 questions

  1. Q3

    June 2025 · Paper 1

    4 marksP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 1, question 3: (a) Evaluate 4 =/ 1 r r 1 giving your answer as a fraction in its lowest terms. [1] (b) Write the sum 1 + + + + in a similar way to the series in part (a). 3 5

  2. Q2

    June 2025 · Paper 2

    2 marksP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 2, question 2: Determine the coefficient of x3 in the expansion of ( 12 2 1 2 )x . [2] +

  3. Q5

    June 2025 · Paper 2

    3 marksP.1 ProofP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 2, question 5: Prove that the sum of the first n positive odd numbers is a square number. [3]

  4. Q1

    June 2025 · Paper 3

    4 marksP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 3, question 1: 1 (a) Find the first three terms in the binomial expansion of ( 1 3 )x . [3] + 2 (b) State the range of values of x for which this expansion is valid. [1]

  5. Q4

    June 2025 · Paper 3

    3 marksP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 3, question 4: The first term of a geometric sequence is 6. 2 . The fourth term is 9 The sequence is infinite. Find the sum of the associated series. [3]

  6. Q7

    June 2025 · Paper 3

    9 marksP.6 Exponentials and logarithmsP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 3, question 7: A group of scientists is studying how a population of insects grows in the laboratory. At the start of the study, there are 48 insects. At the end of the first

  7. Q11

    June 2025 · Paper 3

    5 marksP.2 Algebra and functionsP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 3, question 11: 1 11 The nth term of a sequence is defined by a + , for n H . 1 n = - n 1 1 n 1 1 + - + (a) Prove that every term in the sequence, an, is a rational number. [3]

  8. Q1

    June 2025 · AS Paper 1

    3 marksP.4 Sequences and series
    The whole paper ↗
    June 2025 AS Paper 1, question 1: 1 Expand ( 2 a b ) . You must simplify your answer. [3] -

  9. Q11

    June 2024 · Paper 1

    8 marksP.4 Sequences and seriesP.2 Algebra and functions
    The whole paper ↗
    June 2024 Paper 1, question 11: The first three terms of a geometric sequence are k 5 - , k 2 3 - , k 6 + , where k is a constant. 2 2 - (a) Show that k satisfies the equation k 11 k 10 0 . [3

  10. Q7

    June 2024 · Paper 2

    6 marksP.4 Sequences and series
    The whole paper ↗
    June 2024 Paper 2, question 7: A sequence is defined by the recurrence relation u u 5 with u – 2 . = + = k 1 k 1 + (a) Write down the values of u2, u3, and u4. [1] (b) Explain whether this se

  11. Q10

    June 2024 · Paper 2

    5 marksP.4 Sequences and series
    The whole paper ↗
    June 2024 Paper 2, question 10: (a) Determine the first three terms in ascending powers of x of the binomial expansion 1 )x [4] of ( 8 3 . + 3 (b) State the range of values of x for which this

  12. Q7

    June 2024 · AS Paper 2

    4 marksP.4 Sequences and series
    The whole paper ↗
    June 2024 AS Paper 2, question 7: Determine the coefficient of x5 in the expansion of ( 7 7 3 2 )x . [4] -

  13. Q14

    June 2023 · Paper 1

    6 marksP.4 Sequences and seriesP.6 Exponentials and logarithms
    The whole paper ↗
    June 2023 Paper 1, question 14: (a) Use the laws of logarithms to show that log 200 log 20 is equal to 1. [2] - 10 10 The first three terms of a sequence are log 20 , log 200 , log 2000 . 10 1

  14. Q1

    June 2023 · Paper 2

    3 marksP.4 Sequences and series
    The whole paper ↗
    June 2023 Paper 2, question 1: 1 1 Determine the sum of the infinite geometric series 9 3 1 … [3] - + - + + 3 9

  15. Q7

    June 2023 · Paper 2

    4 marksP.4 Sequences and series
    The whole paper ↗
    June 2023 Paper 2, question 7: The coefficient of x8 in the expansion of ( 12 7 2 x k ) , where k is a positive integer, is 79 200 000. + Determine the value of k. [4]

  16. Q11

    June 2023 · Paper 3

    3 marksP.4 Sequences and seriesP.8 Integration
    The whole paper ↗
    June 2023 Paper 3, question 11: =/ 2 11 (a) Evaluate r . [1] r 1 (b) Show that Euler’s approximate formula, as given in line 13, gives the exact value of 5 =/ 2 r . [2] r 1

  17. Q13

    June 2023 · Paper 3

    4 marksP.4 Sequences and seriesP.8 Integration
    The whole paper ↗
    June 2023 Paper 3, question 13: n =/ 2 13 Prove that Euler’s approximate formula, as given in line 13, when applied to r gives exactly r 1 n n ( 1 2 )( n 1 ) + + . [4] 6 n

  18. Q14

    June 2023 · Paper 3

    3 marksP.8 IntegrationP.4 Sequences and series
    The whole paper ↗
    June 2023 Paper 3, question 14: =/ 1 13 6 n 5 + 14 Show that the expression given in line 33 simplifies to ln n , as given in . + + r 24 12 n n ( 1 ) + r 1 line 34. [3] 6

  19. Q15

    June 2023 · Paper 3

    2 marksP.4 Sequences and series
    The whole paper ↗
    June 2023 Paper 3, question 15: =/ . 1 15 The expression given in line 34 is used to calculate r r 1 Show that the error in the result is less than 1.5% of the true value. [2]

  20. Q4

    June 2022 · Paper 1

    4 marksP.4 Sequences and series
    The whole paper ↗
    June 2022 Paper 1, question 4: Using an appropriate expansion show that, for sufficiently small values of x, x 1 - 1 1 7 2 x x . [4] . - + 4 2 16 2 ( 2 )x +
All 40 questions on Sequences and series, as a list
  1. June 2025, Paper 1, Q3 (4 marks): (a) Evaluate 4 =/ 1 r r 1 giving your answer as a fraction in its lowest terms. [1] (b) Write the sum 1 + + + + in a similar way to the series in part (a). 3 5
  2. June 2025, Paper 2, Q2 (2 marks): Determine the coefficient of x3 in the expansion of ( 12 2 1 2 )x . [2] +
  3. June 2025, Paper 2, Q5 (3 marks): Prove that the sum of the first n positive odd numbers is a square number. [3]
  4. June 2025, Paper 3, Q1 (4 marks): 1 (a) Find the first three terms in the binomial expansion of ( 1 3 )x . [3] + 2 (b) State the range of values of x for which this expansion is valid. [1]
  5. June 2025, Paper 3, Q4 (3 marks): The first term of a geometric sequence is 6. 2 . The fourth term is 9 The sequence is infinite. Find the sum of the associated series. [3]
  6. June 2025, Paper 3, Q7 (9 marks): A group of scientists is studying how a population of insects grows in the laboratory. At the start of the study, there are 48 insects. At the end of the first
  7. June 2025, Paper 3, Q11 (5 marks): 1 11 The nth term of a sequence is defined by a + , for n H . 1 n = - n 1 1 n 1 1 + - + (a) Prove that every term in the sequence, an, is a rational number. [3]
  8. June 2025, AS Paper 1, Q1 (3 marks): 1 Expand ( 2 a b ) . You must simplify your answer. [3] -
  9. June 2024, Paper 1, Q11 (8 marks): The first three terms of a geometric sequence are k 5 - , k 2 3 - , k 6 + , where k is a constant. 2 2 - (a) Show that k satisfies the equation k 11 k 10 0 . [3
  10. June 2024, Paper 2, Q7 (6 marks): A sequence is defined by the recurrence relation u u 5 with u – 2 . = + = k 1 k 1 + (a) Write down the values of u2, u3, and u4. [1] (b) Explain whether this se
  11. June 2024, Paper 2, Q10 (5 marks): (a) Determine the first three terms in ascending powers of x of the binomial expansion 1 )x [4] of ( 8 3 . + 3 (b) State the range of values of x for which this
  12. June 2024, AS Paper 2, Q7 (4 marks): Determine the coefficient of x5 in the expansion of ( 7 7 3 2 )x . [4] -
  13. June 2023, Paper 1, Q14 (6 marks): (a) Use the laws of logarithms to show that log 200 log 20 is equal to 1. [2] - 10 10 The first three terms of a sequence are log 20 , log 200 , log 2000 . 10 1
  14. June 2023, Paper 2, Q1 (3 marks): 1 1 Determine the sum of the infinite geometric series 9 3 1 … [3] - + - + + 3 9
  15. June 2023, Paper 2, Q7 (4 marks): The coefficient of x8 in the expansion of ( 12 7 2 x k ) , where k is a positive integer, is 79 200 000. + Determine the value of k. [4]
  16. June 2023, Paper 3, Q11 (3 marks): =/ 2 11 (a) Evaluate r . [1] r 1 (b) Show that Euler’s approximate formula, as given in line 13, gives the exact value of 5 =/ 2 r . [2] r 1
  17. June 2023, Paper 3, Q13 (4 marks): n =/ 2 13 Prove that Euler’s approximate formula, as given in line 13, when applied to r gives exactly r 1 n n ( 1 2 )( n 1 ) + + . [4] 6 n
  18. June 2023, Paper 3, Q14 (3 marks): =/ 1 13 6 n 5 + 14 Show that the expression given in line 33 simplifies to ln n , as given in . + + r 24 12 n n ( 1 ) + r 1 line 34. [3] 6
  19. June 2023, Paper 3, Q15 (2 marks): =/ . 1 15 The expression given in line 34 is used to calculate r r 1 Show that the error in the result is less than 1.5% of the true value. [2]
  20. June 2022, Paper 1, Q4 (4 marks): Using an appropriate expansion show that, for sufficiently small values of x, x 1 - 1 1 7 2 x x . [4] . - + 4 2 16 2 ( 2 )x +
  21. June 2022, Paper 2, Q2 (2 marks): Find the sum of the infinite series 50 + 25 + 12.5 + 6.25 + … . [2]
  22. June 2022, Paper 3, Q3 (4 marks): n 3 An infinite sequence a , a , a 3 f is defined by a , + , for all positive integers n. n = n 1 1 2 (a) Find the limit of the sequence. [1] (b) Prove that thi
  23. June 2022, Paper 3, Q7 (12 marks): 7 A student is trying to find the binomial expansion of 1 x . - 3 6 x x She gets the first three terms as 1 . - + 2 8 3 3 6 x x x 3 She draws the graphs of the
  24. June 2022, AS Paper 1, Q2 (3 marks): 100 ! . 2 (a) Determine the value of [2] 98 ! (b) Find the coefficient of x98 in the expansion of ( 100 1 x ) . [1] +
  25. November 2021, Paper 1, Q4 (6 marks): (a) The first four terms of a sequence are 2, 3, 0, 3 and the subsequent terms are given by a a . 4 = k k + (i) State what type of sequence this is. [1] 200/ (i
  26. November 2021, Paper 2, Q15 (11 marks): =/ r 1 15 (a) Show that 0 99 . - 0 01 . 1 . [3] = # r 1 Kofi is a very good table tennis player. Layla is determined to beat him. Every week they play one match
  27. November 2021, Paper 3, Q6 (4 marks): In this question you must show detailed reasoning. 3 1 =/ Show that 1 . [4] = r r 1 + + r 1
  28. November 2021, AS Paper 2, Q1 (3 marks): Find the coefficient of x4 in the expansion of ( )x 1 1 3 . [3] +
  29. November 2020, Paper 2, Q5 (3 marks): The first n terms of an arithmetic series are 17 + 28 + 39 + … + 281 + 292. (a) Find the value of n. [1] (b) Find the sum of these n terms. [2]
  30. November 2020, Paper 2, Q6 (4 marks): 6 (a) Find the first three terms in ascending powers of x of the binomial expansion of ( 1 4 )x . [3] + 2 (b) State the range of values of x for which this expa
  31. November 2020, Paper 3, Q1 (2 marks): =/ r 1 Find the value of 2 ( r 1 ) . [2] - r 1
  32. November 2020, Paper 3, Q3 (3 marks): A particular phone battery will last 10 hours when it is first used. Every time it is recharged, it will only last 98% of its previous time. Find the maximum to
  33. June 2019, Paper 1, Q3 (5 marks): - , where a is a non-zero constant. In the binomial 3 3 The function ( )x f is given by ( ) f x ( 1 ax ) = - , the coefficients of x and x2 are equal. expansion
  34. June 2019, Paper 1, Q8 (7 marks): An arithmetic series has first term 9300 and 10th term 3900. (a) Show that the 20th term of the series is negative. [3] (b) The sum of the first n terms is deno
  35. June 2019, Paper 2, Q13 (7 marks): The population of Melchester is 185 207. During a nationwide flu epidemic the number of new cases in Melchester are recorded each day. The results from the firs
  36. June 2018, Paper 1, Q6 (6 marks): Aleela and Baraka are saving to buy a car. Aleela saves £50 in the first month. She increases the amount she saves by £20 each month. (i) Calculate how much she
  37. June 2018, Paper 1, Q13 (12 marks): 8 3 13 The function f (x) is defined by ( ) f x 27 x . Jenny uses her scientific calculator to create a table of = - values for f (x) and f x ( ) . l x f x ( )
  38. June 2018, Paper 3, Q6 (2 marks): c m . 1 2 6 Find the constant term in the expansion of x [2] + x
  39. June 2018, Paper 3, Q13 (3 marks): Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three [3] consecutive terms of this sequence, the middle one
  40. June 2018, AS Paper 1, Q2 (4 marks): ^ h . 3 2 x [4] Find the binomial expansion of 3 2 -

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