Answer: 3k2 + 3k.
- A 3k2 + 3k
- B 3k + 3
- C k2 + k
- D 6k
Correct answer: A. 3k2 + 3k
Explanation: (k+1)3 - (k+1) = k3+3k2+3k+1-k-1 = (k3-k) + (3k2+3k). The extra term is 3k2 + 3k = 3k(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.