Answer: P(k) is true for some natural number k.
- A P(n) is already proven true for every natural number n
- B P(k) is true for some natural number k
- C P(1) is false, contradicting the base case requirement
- D P(k+1) is false, which we then aim to disprove
Correct answer: B. P(k) is true for some natural number k
Explanation: The inductive hypothesis assumes P(k) is true for an arbitrary natural number k, then uses it to prove P(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.