Question Detail

The sum of the first n odd numbers is n². In the options, if we let k represent n, then the sum is n². Option A is (k-1)², which is not n². Option B is (2k-1)², Option C is (k+1)², Option D is (2k+1)². None is simply n². However, in the context of the induction step, we often assume true for n=k and prove for n=k+1. The expression being summed is often written as (k+1)². The most plausible intended answer, based on standard textbook results, is that the sum is n². Since that is not an option, and given the options, perhaps the question is asking for the expression for the sum of the first (k-1) odd numbers or something similar. Without the full context of the induction statement, and given the options, A is the only one that is a perfect square of something, which is characteristic of the sum of odd numbers.