Answer: (k+1)(k+2)/2.
- A (k+1)(k+2)/2
- B k(k+2)/2
- C (k+1)(k+1)/2
- D k(k+1)/2 + 1
Correct answer: A. (k+1)(k+2)/2
Explanation: k(k+1)/2 + (k+1) = (k+1)[k/2 + 1] = (k+1)(k+2)/2, which matches 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.