$(10$ pts$)[$A simplification of Exercise $36,$ Page $538]$ The Josephus survival number f$($n$)$ for n in Z$^(+)$ is defined recursively by f$(1)=1$ and

f$($n$)=2$f$([$n$/2])-(-1)^($n$)$

for n$>=2.$ Prove by induction that

f$($n$)=2($n$-2^(|$logn$|))+1$