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

The 20 questions

1. What must be proven first in a standard proof by mathematical induction?

   • Inductive step
   • Base case
   • Algebraic identity
   • Inductive hypothesis

2. In proof by induction, what is the term for assuming P(k) is true?

   • Inductive hypothesis
   • Base case
   • Inductive conclusion
   • Deductive step

3. What is shown in the inductive step of a proof by induction?

   • P(k+1) is true
   • P(1) is true
   • P(n) is constant
   • P(k) is zero

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. For sum r(r+1) = n(n+1)(n+2)/3, what value do LHS and RHS give when n = 1?

   • Both equal 6
   • Both equal 2
   • Both equal 1
   • Both equal 3

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. When using induction to prove a statement for all positive integers n, what is the base case?

    • n = 1
    • n = k
    • n = k + 1
    • n = 0

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

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