Question Detail
Question
Prove by induction that log(a) + log(a²) + log(a³) + ... + log(a^n) = n(n+1)/2 × log(a) for n ≥ 1.
Correct Answer
Option A is the correct answer.
Detailed Explanation
log(a) + log(a²) + ... + log(a^n) = log(a×a²×...×a^n)=log(a^{1+2+...+n})=log(a^{n(n+1)/2})=n(n+1)/2 × log(a). Base case n=1 ✔. Inductive step follows similarly.
Hint
Use log properties and sum of first n integers.