1.1 Proof by mathematical induction Quiz: Pearson Edexcel Further Maths, Unit 1

The 20 questions

1. In a proof by mathematical induction, what is the purpose of the base case?

   • To show the statement holds for the first value of n, starting the chain of implications
   • To assume the statement is false and derive a contradiction for the first value
   • To show the statement holds only for n = 0 and never for any larger value of n
   • To show the statement holds for every integer at once without any further argument

2. What is the name given to the assumption that the statement is true for n = k in an induction proof?

   • The contrapositive
   • The converse statement
   • The inductive hypothesis
   • The base case

3. Which statement is the correct pattern for the sum of the first n odd numbers?

   • 1 + 3 + 5 + ... + (2n - 1) = n(n + 1)
   • 1 + 3 + 5 + ... + (2n - 1) = n^2
   • 1 + 3 + 5 + ... + (2n - 1) = n^3
   • 1 + 3 + 5 + ... + (2n - 1) = 2n^2

4. A statement is to be proved by induction for all integers n >= 2. At which value must the base case be checked?

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

5. For the result sum of r^2 from r = 1 to n equal to n(n + 1)(2n + 1)/6, what is the value of both sides when n = 1?

   • Both sides equal 6
   • Both sides equal 2
   • Both sides equal 1
   • The left side is 1 and the right side is 2

6. Why is checking several specific values of n not, by itself, a valid proof that a statement holds for all positive integers?

   • Checking values proves the statement is false whenever the check succeeds
   • Checking values only works for statements about negative integers, which are excluded
   • Checking values is always a valid proof for any statement about positive integers
   • Finitely many checks cannot cover infinitely many integers, so a general argument linking each case to the next is needed

7. What is the value of sum of r(r + 1) from r = 1 to 4?

   • 36
   • 30
   • 40
   • 50

8. For sum of r(r + 1) from r = 1 to n equal to n(n + 1)(n + 2)/3, what must be added to the sum when moving from n = k to n = k + 1?

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

9. For M = [[3, -4], [1, -1]], what is the (1,2) entry of M squared?

   • 8
   • -3
   • -8
   • -4

10. Which expression is divisible by 6 for every positive integer n?

    • 7^n + 1
    • 7^n - 3
    • 7^n - 1
    • 7^n - 2

11. In proving 6 divides 7^n - 1 by induction, which expression for 7^(k+1) - 1 shows divisibility by 6 once 6 divides 7^k - 1 is assumed?

    • 7(7^k + 1) - 6
    • 6 x 7^k + 1
    • 7(7^k - 1) - 6
    • 7(7^k - 1) + 6

12. What is the base case check for the inequality 2^n >= n + 1 at n = 0?

    • 1 >= 1, which holds
    • 0 >= 1, which fails
    • 1 > 1, which is false so the base case fails
    • 2 >= 1, which holds

13. In the inductive step for 2^n >= n + 1, which chain completes the argument from 2^k >= k + 1?

    • 2^(k+1) = 2 x 2^k >= 2k + 2 >= k + 2
    • 2^(k+1) = 2^k + 1 >= k + 2
    • 2^(k+1) >= k + 2 directly, without using the inductive hypothesis
    • 2^(k+1) >= 2k, so the result holds only for k >= 1 and needs no further argument

14. What is the sum of the cubes from r = 1 to 3?

    • 45
    • 36
    • 24
    • 27

15. If the sum of cubes from r = 1 to n equals n^2(n + 1)^2/4, what is the sum when n = 4?

    • 100
    • 64
    • 120
    • 90

16. For the statement that k^3 + 2k is divisible by 3, which expansion correctly shows (k+1)^3 + 2(k+1) = (k^3 + 2k) + 3(k^2 + k + 1)?

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

17. In the matrix induction for M^n = [[2n+1, -4n], [n, 1-2n]], which expression gives the (2,2) entry of M^(k+1) = M x M^k?

    • -(4k) - 2k, which simplifies to -6k
    • -4k + (1 - 2k), which simplifies to 1 - 6k
    • 1 - 2k - 1, which simplifies to -2k
    • -4k - (1 - 2k), which simplifies to -2k - 1 = 1 - 2(k+1)

18. Why is verifying only the base case n = 1 insufficient to prove a statement by induction?

    • It does not by itself link the case n = k to n = k + 1, so no conclusion for larger n follows
    • It must be replaced by a check at n = 0, which induction never allows
    • It proves only that the statement is false, so it cannot support a positive conclusion
    • It is sufficient only when the statement involves matrices, not when it involves sums

19. A statement P(n) is true for n = 1, and P(k) true implies P(k + 2) true for every positive integer k. Which values does this establish?

    • All odd positive integers only
    • Only n = 1 and n = 2
    • All positive integers
    • All even positive integers

20. In strong induction, what assumption is made in the inductive step?

    • That the statement is false for k + 1
    • That the statement holds for k only and fails for every smaller value
    • That the statement holds for all values from the base case up to k
    • That n is an even integer

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