Skip to content
Try on iPad

Proof by induction: OCR A‑Level Further Maths questions

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

Find questions648

648 questions

  1. Q8

    June 2025 · Pure Core 2

    12 marksCP.1 Proof by inductionCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 Pure Core 2, question 8: A function f (x) is defined by f (x) = xsinh 2x. 2 n 0 d f d f n (a) Prove by induction that 4 ( x sinh 2 x n cosh 2 x ) for n 0 where d is defined as H = + 2 n

  2. Q7

    June 2025 · AS Pure Core

    5 marksCP.1 Proof by induction
    The whole paper ↗
    June 2025 AS Pure Core, question 7: 20n 7 n for all integers n [5] Prove by induction that ! 2 52 . H J N 5 a 5 1 +

  3. Q5

    June 2025 · AS Additional Pure Mathematics

    5 marksCP.1 Proof by induction
    The whole paper ↗
    June 2025 AS Additional Pure Mathematics, question 5: n n 5 f n for all positive integers n. Let ( ) 3 4 = + f n is a multiple of 7 for all odd integers n [5] Prove by induction that ( ) H . 1

  4. Q8

    June 2024 · Pure Core 1

    5 marksCP.1 Proof by induction
    The whole paper ↗
    June 2024 Pure Core 1, question 8: n n 8 - is divisible by 3, for all integers n [5] Prove by induction that 11 7 13 1 0 . - # H

  5. Q6

    June 2024 · AS Pure Core

    5 marksCP.1 Proof by induction
    The whole paper ↗
    June 2024 AS Pure Core, question 6: J N 1 a KK OO where a is a constant. 6 You are given that A = 0 1 L P J N 1 an n = KK OO for all integers n Prove by induction that A H . 1 [5] 0 1 L P

  6. Q4

    June 2024 · AS Additional Pure Mathematics

    5 marksFP.9 Further sequences and seriesCP.1 Proof by induction
    The whole paper ↗
    June 2024 AS Additional Pure Mathematics, question 4: " ,, where n 4 The first five terms of the Fibonacci sequence, Fn H , are 1 F 1 , F 1 , F 2 , F = and F 3 5 = . 5 = = = 1 2 3 4 (a) Use the recurrence definitio

  7. Q9

    June 2023 · Pure Core 2

    9 marksCP.1 Proof by inductionCP.4 Further algebra and functions
    The whole paper ↗
    June 2023 Pure Core 2, question 9: A function is defined by y f ( ) t where ( ) f t ln ( 1 at ) and a is a constant. = = + 2 3 4 n d , t y d y d y d y d y (a) By considering t , t and t , make a

  8. Q6

    June 2023 · AS Pure Core

    6 marksCP.1 Proof by induction
    The whole paper ↗
    June 2023 AS Pure Core, question 6: n 6 Prove by induction that 4 8 66 is divisible by 14 for all integers n 0 . [6] + H #

  9. Q6

    June 2022 · Pure Core 1

    6 marksCP.1 Proof by induction
    The whole paper ↗
    June 2022 Pure Core 1, question 6: Let y x cosh x . = 2 n 1 d - y Prove by induction that, for all integers n H , 1 x sinh x ( 2 n 1 ) cosh x . [6] = + - 2 n 1 d x -

  10. Q3

    June 2022 · Additional Pure Mathematics

    6 marksFP.9 Further sequences and seriesCP.1 Proof by induction
    The whole paper ↗
    June 2022 Additional Pure Mathematics, question 3: ^ h plays a significant role in the sequence of Fibonacci 1 3 The irrational number 1 5 z = + 2 numbers given by F 0 , F 1 = and 1 F F F - for n H . 1 0 = = + n

  11. Q4

    June 2022 · AS Pure Core

    5 marksCP.1 Proof by induction
    The whole paper ↗
    June 2022 AS Pure Core, question 4: n 2 4 Prove that 3 10 n for all integers n 4 . [5] H

  12. Q9

    November 2021 · Pure Core 2

    6 marksCP.1 Proof by inductionCP.3 Matrices
    The whole paper ↗
    November 2021 Pure Core 2, question 9: J N 2 3 KK OO. 9 The matrix A is given by A = 0 2 L P (a) By considering A, A2, A3 and A4 make a conjecture about the form of the matrix An in terms of n for n

  13. Q7

    November 2021 · AS Pure Core

    5 marksCP.1 Proof by induction
    The whole paper ↗
    November 2021 AS Pure Core, question 7: - n n 7 Prove that 2 3 is divisible by 5 for all integers n H . 1 [5]

  14. Q3

    November 2021 · AS Additional Pure Mathematics

    6 marksFP.9 Further sequences and seriesCP.1 Proof by induction
    The whole paper ↗
    November 2021 AS Additional Pure Mathematics, question 3: For positive integers n, the sequence of Fibonacci numbers, { Fn , starts with the terms } F 1 = , F 1 2 = , F 1 2 , … and is given by the recurrence relation F

  15. Q7

    November 2020 · Pure Core 1

    5 marksCP.1 Proof by induction
    The whole paper ↗
    November 2020 Pure Core 1, question 7: Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9. [5]

  16. Q8

    November 2020 · Additional Pure Mathematics

    12 marksFP.9 Further sequences and seriesCP.1 Proof by induction
    The whole paper ↗
    November 2020 Additional Pure Mathematics, question 8: u 3 " , of positive real numbers is defined by u + n 8 The sequence un 1 = and u for n H . 1 1 1 = u 2 n + + n 2 (a) Prove by induction that u 3 1 0 for all pos

  17. Q6

    November 2020 · AS Pure Core

    5 marksCP.1 Proof by induction
    The whole paper ↗
    November 2020 AS Pure Core, question 6: n 2 6 Prove that ! n 2 for all integers n H 9. [5]

  18. Q8

    June 2019 · AS Pure Core

    6 marksCP.1 Proof by induction
    The whole paper ↗
    June 2019 AS Pure Core, question 8: In this question you must show detailed reasoning. M is the matrix 1 6 f p. 0 2 n 1 f 1 3 2 ( + 2 ) p, for any positive integer n. - n Prove that M [6] = n 0 2

  19. Q7

    June 2018 · AS Pure Core

    6 marksCP.1 Proof by induction
    The whole paper ↗
    June 2018 AS Pure Core, question 7: n 1 n 7 Prove by induction that 2 + 5 9 is divisible by 7 for all integers n H 1. [6] + #
All 19 questions on Proof by induction, as a list
  1. June 2025, Pure Core 2, Q8 (12 marks): A function f (x) is defined by f (x) = xsinh 2x. 2 n 0 d f d f n (a) Prove by induction that 4 ( x sinh 2 x n cosh 2 x ) for n 0 where d is defined as H = + 2 n
  2. June 2025, AS Pure Core, Q7 (5 marks): 20n 7 n for all integers n [5] Prove by induction that ! 2 52 . H J N 5 a 5 1 +
  3. June 2025, AS Additional Pure Mathematics, Q5 (5 marks): n n 5 f n for all positive integers n. Let ( ) 3 4 = + f n is a multiple of 7 for all odd integers n [5] Prove by induction that ( ) H . 1
  4. June 2024, Pure Core 1, Q8 (5 marks): n n 8 - is divisible by 3, for all integers n [5] Prove by induction that 11 7 13 1 0 . - # H
  5. June 2024, AS Pure Core, Q6 (5 marks): J N 1 a KK OO where a is a constant. 6 You are given that A = 0 1 L P J N 1 an n = KK OO for all integers n Prove by induction that A H . 1 [5] 0 1 L P
  6. June 2024, AS Additional Pure Mathematics, Q4 (5 marks): " ,, where n 4 The first five terms of the Fibonacci sequence, Fn H , are 1 F 1 , F 1 , F 2 , F = and F 3 5 = . 5 = = = 1 2 3 4 (a) Use the recurrence definitio
  7. June 2023, Pure Core 2, Q9 (9 marks): A function is defined by y f ( ) t where ( ) f t ln ( 1 at ) and a is a constant. = = + 2 3 4 n d , t y d y d y d y d y (a) By considering t , t and t , make a
  8. June 2023, AS Pure Core, Q6 (6 marks): n 6 Prove by induction that 4 8 66 is divisible by 14 for all integers n 0 . [6] + H #
  9. June 2022, Pure Core 1, Q6 (6 marks): Let y x cosh x . = 2 n 1 d - y Prove by induction that, for all integers n H , 1 x sinh x ( 2 n 1 ) cosh x . [6] = + - 2 n 1 d x -
  10. June 2022, Additional Pure Mathematics, Q3 (6 marks): ^ h plays a significant role in the sequence of Fibonacci 1 3 The irrational number 1 5 z = + 2 numbers given by F 0 , F 1 = and 1 F F F - for n H . 1 0 = = + n
  11. June 2022, AS Pure Core, Q4 (5 marks): n 2 4 Prove that 3 10 n for all integers n 4 . [5] H
  12. November 2021, Pure Core 2, Q9 (6 marks): J N 2 3 KK OO. 9 The matrix A is given by A = 0 2 L P (a) By considering A, A2, A3 and A4 make a conjecture about the form of the matrix An in terms of n for n
  13. November 2021, AS Pure Core, Q7 (5 marks): - n n 7 Prove that 2 3 is divisible by 5 for all integers n H . 1 [5]
  14. November 2021, AS Additional Pure Mathematics, Q3 (6 marks): For positive integers n, the sequence of Fibonacci numbers, { Fn , starts with the terms } F 1 = , F 1 2 = , F 1 2 , … and is given by the recurrence relation F
  15. November 2020, Pure Core 1, Q7 (5 marks): Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9. [5]
  16. November 2020, Additional Pure Mathematics, Q8 (12 marks): u 3 " , of positive real numbers is defined by u + n 8 The sequence un 1 = and u for n H . 1 1 1 = u 2 n + + n 2 (a) Prove by induction that u 3 1 0 for all pos
  17. November 2020, AS Pure Core, Q6 (5 marks): n 2 6 Prove that ! n 2 for all integers n H 9. [5]
  18. June 2019, AS Pure Core, Q8 (6 marks): In this question you must show detailed reasoning. M is the matrix 1 6 f p. 0 2 n 1 f 1 3 2 ( + 2 ) p, for any positive integer n. - n Prove that M [6] = n 0 2
  19. June 2018, AS Pure Core, Q7 (6 marks): n 1 n 7 Prove by induction that 2 + 5 9 is divisible by 7 for all integers n H 1. [6] + #

Questions, mark schemes and examiner’s reports are © OCR, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every OCR A-Level Further Maths paper.

Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.