Sequences And Series

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