Integrals I: Feynman's Trick · Superchallenge problem

Define $f(1) = 1$ and $f(n) = f(\lfloor n/2\rfloor) + 1$ for $n \ge 2$. Strong induction proves $f(n) = \lfloor\log_2 n\rfloor + 1$. Compute $\displaystyle\sum_{n=1}^{100} f(n)$.