4.01 Proof by induction Quiz: OCR Further Maths, Unit 1

The 20 questions

1. What is the initial step in a proof by induction for positive integers?

   • Conclusion
   • Inductive step
   • Inductive hypothesis
   • Base case

2. What term describes the assumption that a statement holds for n = k?

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

3. Which statement must be proved true during the inductive step of a proof?

   • Statement for k - 1
   • Statement for 2k
   • Statement for 1
   • Statement for k + 1

4. What is the correct formula for the sum of the first n positive integers?

   • n^2 / 2
   • n(n + 1)
   • n(n - 1) / 2
   • n(n + 1) / 2

5. Which formula gives the sum of the first n square integers?

   • n(2n+1) / 3
   • n(n+1)(n+2) / 6
   • n^2(n+1)^2 / 4
   • n(n+1)(2n+1) / 6

6. What is the simplified expression for the sum of the first n cubes?

   • n^3(n + 1)^3 / 8
   • n(n + 1) / 2
   • [n(n + 1) / 2]^2
   • n^2(n + 1) / 4

7. If matrix A has top row 2, 0 and bottom row 0, 2, what is A^n?

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

8. If matrix M has top row 1, 3 and bottom row 0, 1, what is M^n?

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

9. When testing the base case for 5^n - 1 being divisible by 4, what is f(1)?

   • 5
   • 1
   • 0
   • 4

10. For f(n) = 3^(2n) - 1, what is the simplified difference f(k+1) - f(k)?

    • 2 * 3^(2k)
    • 9 * 3^(2k)
    • 3^(2k)
    • 8 * 3^(2k)

11. If a property is to be proved by induction for all n >= 3, what base case is evaluated?

    • n = 0
    • n = 1
    • n = 2
    • n = 3

12. In a proof by matrix induction, how is A^(k+1) rewritten for the inductive step?

    • A^k * A^k
    • A^k * A
    • (A^k)^2
    • A^k + A

13. What is the closed-form sum of the geometric series from r = 1 to n of 2^r?

    • 2^n - 2
    • 2^(n+1) - 2
    • 2^n - 1
    • 2^(n+1) - 1

14. Which non-zero integer cleanly divides n^3 - n for all positive integers n?

    • 8
    • 6
    • 4
    • 12

15. What base case value of n is tested first to prove 2^n > n^2 for n >= 5?

    • 1
    • 5
    • 4
    • 2

16. What essential element must conclude any valid proof by mathematical induction?

    • Counterexample
    • Integration constant
    • Algebraic expansion
    • Concluding statement

17. For matrix A with rows [1, 0] and [2, 1], what is the bottom-left entry of A^n?

    • 2n
    • n^2
    • 2^n
    • 2

18. If f(n) = 7^n + 4^n, what is the correct expression for f(k+1)?

    • 11^(k+1)
    • 7^(k+1) + 4^(k+1)
    • 7(7^k + 4^k)
    • 7^k + 4^k + 1

19. What term is added to both sides of the inductive hypothesis for the sum of integers?

    • k
    • k + 1
    • 2k + 1
    • k^2

20. Why is mathematical induction formally classified as a method of deductive proof?

    • Logical necessity
    • Statistical inference
    • Empirical observation
    • Approximate reasoning

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