Lesson A1

A1 Proof by mathematical induction Quiz: AQA Further Maths, Unit 2

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

In partnership with Revision Ninja

Lesson A1, Proof by mathematical induction: 20 multiple choice questions for the AQA Further Maths (7367), Unit 2: Compulsory content, written with Revision Ninja.

Host it live on the board and students join with a game code on their own devices, or revise alone with Free Play. The answers are revealed in the game.

Host this setFree Play

The 20 questions

  1. In a proof by induction, what must be shown in the base case?

    • The statement is true for all n at once
    • The statement holds for the first value of n, usually n = 1
    • The statement is false for n = 1
    • The statement holds for n = 0 only, with no other checks

  2. Assuming a statement P(k) is true for some integer k is called what?

    • The inductive hypothesis
    • The base case, which checks the statement for the first value of n only
    • The contradiction, which assumes the statement is false for some value of n
    • The converse, which reverses the implication used in the inductive step

  3. After assuming P(k) is true, what must the proof show to complete the inductive step?

    • P(k + 1) follows from P(k)
    • P(k - 1) follows from P(k)
    • P(k) is false for large k
    • P(2k) follows from P(k)

  4. For the sum of r from 1 to k, adding k + 1 to k(k + 1)/2 gives which expression, as required in the inductive step?

    • k(k + 2)/2
    • (k + 1)(k + 2)/2
    • k(k + 1)/2 + k
    • (k + 1)^2/2

  5. What is the sum of the squares 1^2 + 2^2 + 3^2?

    • 12
    • 16
    • 14
    • 36

  6. What is the sum of the cubes 1^3 + 2^3 + 3^3 + 4^3?

    • 96
    • 120
    • 100
    • 64

  7. Show that n(n + 1)(n + 2)/3 gives the correct sum when n = 1 for the series of r(r + 1). Which pair of values confirms the base case?

    • For n = 2: LHS = 6 and RHS = 8
    • For n = 1: LHS = 2 and RHS = 2
    • For n = 1: LHS = 1 and RHS = 2
    • For n = 1: LHS = 4 and RHS = 4

  8. For which n >= 1 does 2^n > n^2 fail to hold?

    • n = 4 only
    • n = 1 only
    • n = 2, 3 and 4
    • n = 5 and 6 only

  9. In the inductive step for sum of r(r + 1) = n(n + 1)(n + 2)/3, what does (k + 1)(k + 2)(k + 3)/3 - k(k + 1)(k + 2)/3 simplify to?

    • (k + 2)(k + 3)
    • k(k + 1)
    • (k + 1)(k + 3)
    • (k + 1)(k + 2)

  10. Prove that 7 divides 8^n - 1. If 8^k - 1 = 7m, what is 8^(k+1) - 1 in terms of m?

    • 7(8m - 1)
    • 8(7m) - 1
    • 7m + 7
    • 7(8m + 1)

  11. For the matrix M = [[1, 1], [0, 1]], what is M^(k+1) written as a product in the inductive step when M^k = [[1, k], [0, 1]]?

    • [[1, k], [0, 1]]
    • [[k + 1, 1], [0, k + 1]]
    • [[1, 2k], [0, 1]]
    • [[1, k + 1], [0, 1]]

  12. A sequence has u1 = 2 and u(n+1) = 3u(n) - 1. Which closed form gives u(n)?

    • u(n) = 3^n + 1
    • u(n) = 2 x 3^(n-1)
    • u(n) = (3^(n-1) + 1)/2
    • u(n) = (3^n + 1)/2

  13. What is the nth derivative of x e^x?

    • x e^(nx)
    • (x + n)e^x
    • n x e^x
    • (x + 1)e^(nx)

  14. For the inductive step of the cubic sum (k(k + 1)/2)^2 + (k + 1)^3, which common factor is taken out?

    • k(k + 1)
    • (k + 2)^2
    • (k + 1)^2
    • (k + 1)/2

  15. The claim n^2 + n + 41 is prime for all n >= 1 holds for n = 1 to 40. Which statement is correct?

    • The claim fails at n = 41, since 41^2 + 41 + 41 = 41 x 43 is composite
    • The claim holds for all n because 41 is prime
    • The claim is true for n up to 40, so it must be true always
    • The claim is proved since the first 40 cases hold

  16. To prove that 2 divides n^2 + n by induction, which expression equals (k + 1)^2 + (k + 1)?

    • (k^2 + k) + 2k
    • (k^2 + k) + 2(k + 1)
    • k^2 + 2k + 1
    • (k^2 + k) + (k + 1)

  17. For a sequence defined by u(n+2) = u(n+1) + u(n), which base cases are needed to prove a statement about all n by induction?

    • n = 0 and n = 1 only
    • n = 1 only
    • n = 1 and n = 2
    • n = 2 and n = 3

  18. For M = [[2, 0], [0, 3]], what is M^n?

    • [[n^2, 0], [0, n^3]]
    • [[2n, 0], [0, 3n]]
    • [[2^n, 0], [0, 3^n]]
    • [[2^n, 1], [0, 3^n]]

  19. For which odd n is 10^n + 1 divisible by 11?

    • All even n
    • All odd n
    • No value of n
    • Only n = 1

  20. What is the sum of r(r + 1) for r from 1 to 5?

    • 75
    • 80
    • 70
    • 60

All AQA Further Maths quizzes