Answer: [(k+1)(k+2)/2]2.
- A [(k+1)(k+2)/2]2
- B [k(k+2)/2]2
- C (k+1)2 (k+2)2
- D k2(k+1)2/4 + (k+1)
Correct answer: A. [(k+1)(k+2)/2]2
Explanation: Factor out (k+1)2: [k(k+1)/2]2 + (k+1)3 = (k+1)2 [k2/4 + (k+1)] = (k+1)2 [(k2+4k+4)/4] = (k+1)2 (k+2)2/4 = [(k+1)(k+2)/2]2, matching the formula for n=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.