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 \(k^{\text{th}\text{~}}\) match are given by \(k{.2}^{n + 1 - k}\), where \(1 \leq k \leq n\). Find \(n\)