Answer: 1+3+5+...+(2k-1) = k 2.
- A 1+3+5+...+(2k-1) = k<sup>2</sup>
- B 1+3+5+...+(2k+1) = k<sup>2</sup>
- C 1+2+...+k = k<sup>2</sup>
- D k = k<sup>2</sup>
Correct answer: A. 1+3+5+...+(2k-1) = k<sup>2</sup>
Explanation: P(k) states that the sum of the first k odd numbers, 1+3+5+...+(2k-1), equals k<sup>2.</sup>
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.