Question Detail
Question
Prove by induction that 1×2⁰ + 2×2¹ + 3×2² + ... + n×2^{n-1} = (n−1)2^n + 1 for n ≥ 1.
Correct Answer
Option A is the correct answer.
Detailed Explanation
Base n=1: 1×2⁰ = 1 = (1−1)2¹ +1 ✔. Assume n=k, then add (k+1)×2^k → factorization yields (k)2^{k+1}+1 ✔.
Hint
Use induction and factor powers of 2.