Question

If total number of runs scored in n matches is $\left(\frac{n+1}{4}\right)(2^{n+1}-n-2)$ where n > 1, and the runs scored
in the $k^{th}$ match are given by k $2^{n+1}$ where $1<k<n$. Find n.

Answer 1

Let $s_n={\sum }_{k=1}^{n}k{.2}^{-k}=2^{n+1}.2[1-\frac{1}{2^n}-\frac{n}{2^{n+1}}]$ $=2[2^{n+1}-2-n]\Rightarrow \frac{n+1}{4}=2\Rightarrow n=7.$