2021 SJI P2 Q6

Using the result

\[\sum_{r=1}^n\left(\frac r{2^r}\right)=2-\frac{n+2}{2^n},\]

show that

\[\sum_{r=1}^n(r-n)(2^{-r}+1)=2\left(1-\frac1{2^n}\right)-\frac12n(n+1).\][7]

Find the positive values of \(n\) such that \[\sum_{r=1}^n(r-n)(2^{-r}+1)>-\ln12.\][3]