Question Detail
Question
Prove by induction that 1×2 + 2×3 + ... + n(n+1) = n(n+1)(n+2)/3 for n ≥ 1.
Correct Answer
Option A is the correct answer.
Detailed Explanation
Base case n=1: 1×2 = 1×2×3/3 = 2 ✔. Assume true for n=k. For n=k+1: sum = k(k+1)(k+2)/3 + (k+1)(k+2) = (k+1)(k+2)(k/3 +1) = (k+1)(k+2)(k+3)/3 ✔.
Hint
Factor out (k+1)(k+2) when adding the next term.