Answer: 2k + 1.
- A 2k - 1
- B 2k + 1
- C k + 1
- D 2k
Correct answer: B. 2k + 1
Explanation: The (k+1)th odd number is 2(k+1) - 1 = 2k + 1. Adding this to k<sup>2</sup> gives k<sup>2</sup> + 2k + 1 = (k+1)<sup>2</sup>, proving 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.