Answer: Assume k(k+1)(k+2) is divisible by 6, then show (k+1)(k+2)(k+3) is divisible by 6.
- A Assume k(k+1)(k+2) is divisible by 6, then show (k+1)(k+2)(k+3) is divisible by 6
- B Assume n itself is divisible by 6, without ever involving consecutive products together
- C Assume just the single term (k+1) is divisible by 6, ignoring the rest
- D Assume k(k+1) is divisible by 3, dropping the third factor from consideration
Correct answer: A. Assume k(k+1)(k+2) is divisible by 6, then show (k+1)(k+2)(k+3) is divisible by 6
Explanation: The inductive hypothesis P(k) assumes the product of three consecutive integers starting at k is divisible by 6; the inductive step must then prove the same for the next set of three consecutive integers starting at k+1.
Mathematical induction works like a row of dominoes: proving the base case P(1) tips the first domino, and proving the inductive step (P(k) ⟹ P(k+1)) guarantees each domino knocks over the next - together these two facts guarantee ALL dominoes fall, without checking each one individually.
Concept context
A proof technique used to establish that a statement is true for every natural number, using a base case and an inductive step.