An induction proof assumes the very statement it is trying to prove holds for n, then proves it for n+1. Why is that not circular reasoning?

C. Because it assumes the statement only for one n, to link consecutive cases.

The inductive step does not prove P(n+1); it proves 'if P(n) then P(n+1)', and assuming a premise to prove a conditional is ordinary practice. That leaves a chain with nothing travelling along it until the base case supplies the first true statement. 'n = n + 1' has a perfectly valid step and no base, and is false everywhere.