Answer: that xᵏ − yᵏ is divisible.
- A that xᵏ − yᵏ is divisible
- B only the given base case
- C a purely lucky guess
- D no earlier known result
Correct answer: A. that xᵏ − yᵏ is divisible
Explanation: x<sup>k+1</sup> − y<sup>k+1</sup> is manipulated to contain xᵏ − yᵏ, which the hypothesis says is divisible by (x − y).
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.