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Proof by induction: OCR MEI A‑Level Further Maths questions

Every OCR MEI A-Level Further Maths exam question on CP.1 Proof by induction, 2018–2025, with its mark scheme and examiner’s report.

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847 questions

  1. Q8

    June 2025 · Core Pure

    7 marksCP.1 Proof by inductionCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Core Pure, question 8: The function ( )x f is defined as ( ) f x ln ( 1 x ) , for x 2- . 1 = + f ( ) n (a) Prove by mathematical induction that the nth derivative of ( ) f x , ( )x ,

  2. Q8

    June 2024 · Core Pure

    10 marksCP.3 MatricesCP.1 Proof by induction
    The whole paper ↗
    June 2024 Core Pure, question 8: (a) Specify fully the transformation T of the plane associated with the matrix M, where = c 1 m and m is a non-zero constant. m M [2] 0 1 (b) (i) Find det M. [1

  3. Q6

    June 2024 · AS Core Pure

    9 marksCP.1 Proof by inductionCP.3 Matrices
    The whole paper ↗
    June 2024 AS Core Pure, question 6: J N 4 9 - KK OO. 6 You are given that M = 1 2 - L P J N n n 1 3 9 + - n = KK OO for all positive integers n. (a) Prove that M [6] n 1 3 n - L P (b) A student th

  4. Q8

    June 2023 · Core Pure

    5 marksCP.1 Proof by induction
    The whole paper ↗
    June 2023 Core Pure, question 8: n n 8 Prove by mathematical induction that 8 3 is divisible by 5 for all positive integers n. [5] -

  5. Q6

    June 2023 · AS Core Pure

    8 marksCP.3 MatricesCP.1 Proof by induction
    The whole paper ↗
    June 2023 AS Core Pure, question 6: J N 2 1 KK OO. 6 The matrix M is 1 0 - L P (a) Calculate M2, M3 and M4. [2] (b) Hence make a conjecture about the matrix Mn. [1] (c) Prove your conjecture. [5]

  6. Q6

    June 2022 · Core Pure

    5 marksCP.1 Proof by inductionCP.3 Matrices
    The whole paper ↗
    June 2022 Core Pure, question 6: J N J N n 2 0 2 0 KK OO KK OO for all positive integers n. 6 Prove by mathematical induction that [5] = n 1 1 1 2 1 - - L P L P 3 Answer all the questions.

  7. Q6

    June 2022 · AS Core Pure

    10 marksCP.4 Further algebra and functionsCP.1 Proof by induction
    The whole paper ↗
    June 2022 AS Core Pure, question 6: n =/ 1 6 (a) Using standard summation formulae, show that r r ( 2 ) n n ( 1 2 )( n 7 ). [4] + = + + 6 r 1 (b) Use induction to prove the result in part (a). [6]

  8. Q7

    November 2021 · Core Pure

    6 marksCP.1 Proof by induction
    The whole paper ↗
    November 2021 Core Pure, question 7: n r/ r n 2 + 7 Prove that 4 for all n H . 1 [6] = - r 1 n 1 2 - 2 - 1 =

  9. Q5

    November 2021 · AS Core Pure

    5 marksCP.1 Proof by induction
    The whole paper ↗
    November 2021 AS Core Pure, question 5: n r/ r n 1 5 Prove by induction that r 2 - 1 ( n 1 2 ) for all positive integers n. [5] = + - # 1 =

  10. Q7

    November 2020 · Core Pure

    6 marksCP.1 Proof by induction
    The whole paper ↗
    November 2020 Core Pure, question 7: n =/ 7 Prove by mathematical induction that ( r r )! ( n 1 )! 1 for all positive integers n. [6] = + - # r 1 J N J N J N J N 0 1 1 2 - -

  11. Q5

    November 2020 · AS Core Pure

    6 marksCP.1 Proof by induction
    The whole paper ↗
    November 2020 AS Core Pure, question 5: You are given that u 1 = and u 5 u 2 n 4 . 1 = + + n n + 2 Prove by induction that u n 3 n + for all positive integers n. 1 [6] = + n

  12. Q9

    June 2019 · Core Pure

    7 marksCP.1 Proof by induction
    The whole paper ↗
    June 2019 Core Pure, question 9: n n 9 Prove by induction that 5 2 11 is divisible by 3 for all positive integers n. [7] + #

  13. Q5

    June 2019 · AS Core Pure

    6 marksCP.1 Proof by induction
    The whole paper ↗
    June 2019 AS Core Pure, question 5: J N n 1 1 1 1 / KK OO 5 Prove by induction that, for all positive integers n, . [6] = - r 2 n 3 3 r 1 L P =

  14. Q8

    June 2018 · AS Core Pure

    6 marksCP.1 Proof by inductionCP.3 Matrices
    The whole paper ↗
    June 2018 AS Core Pure, question 8: c m n c n m for all positive integers n. Prove by induction that 1 1 1 2 1 - 8 [6] = n 0 2 0 2
All 14 questions on Proof by induction, as a list
  1. June 2025, Core Pure, Q8 (7 marks): The function ( )x f is defined as ( ) f x ln ( 1 x ) , for x 2- . 1 = + f ( ) n (a) Prove by mathematical induction that the nth derivative of ( ) f x , ( )x ,
  2. June 2024, Core Pure, Q8 (10 marks): (a) Specify fully the transformation T of the plane associated with the matrix M, where = c 1 m and m is a non-zero constant. m M [2] 0 1 (b) (i) Find det M. [1
  3. June 2024, AS Core Pure, Q6 (9 marks): J N 4 9 - KK OO. 6 You are given that M = 1 2 - L P J N n n 1 3 9 + - n = KK OO for all positive integers n. (a) Prove that M [6] n 1 3 n - L P (b) A student th
  4. June 2023, Core Pure, Q8 (5 marks): n n 8 Prove by mathematical induction that 8 3 is divisible by 5 for all positive integers n. [5] -
  5. June 2023, AS Core Pure, Q6 (8 marks): J N 2 1 KK OO. 6 The matrix M is 1 0 - L P (a) Calculate M2, M3 and M4. [2] (b) Hence make a conjecture about the matrix Mn. [1] (c) Prove your conjecture. [5]
  6. June 2022, Core Pure, Q6 (5 marks): J N J N n 2 0 2 0 KK OO KK OO for all positive integers n. 6 Prove by mathematical induction that [5] = n 1 1 1 2 1 - - L P L P 3 Answer all the questions.
  7. June 2022, AS Core Pure, Q6 (10 marks): n =/ 1 6 (a) Using standard summation formulae, show that r r ( 2 ) n n ( 1 2 )( n 7 ). [4] + = + + 6 r 1 (b) Use induction to prove the result in part (a). [6]
  8. November 2021, Core Pure, Q7 (6 marks): n r/ r n 2 + 7 Prove that 4 for all n H . 1 [6] = - r 1 n 1 2 - 2 - 1 =
  9. November 2021, AS Core Pure, Q5 (5 marks): n r/ r n 1 5 Prove by induction that r 2 - 1 ( n 1 2 ) for all positive integers n. [5] = + - # 1 =
  10. November 2020, Core Pure, Q7 (6 marks): n =/ 7 Prove by mathematical induction that ( r r )! ( n 1 )! 1 for all positive integers n. [6] = + - # r 1 J N J N J N J N 0 1 1 2 - -
  11. November 2020, AS Core Pure, Q5 (6 marks): You are given that u 1 = and u 5 u 2 n 4 . 1 = + + n n + 2 Prove by induction that u n 3 n + for all positive integers n. 1 [6] = + n
  12. June 2019, Core Pure, Q9 (7 marks): n n 9 Prove by induction that 5 2 11 is divisible by 3 for all positive integers n. [7] + #
  13. June 2019, AS Core Pure, Q5 (6 marks): J N n 1 1 1 1 / KK OO 5 Prove by induction that, for all positive integers n, . [6] = - r 2 n 3 3 r 1 L P =
  14. June 2018, AS Core Pure, Q8 (6 marks): c m n c n m for all positive integers n. Prove by induction that 1 1 1 2 1 - 8 [6] = n 0 2 0 2

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