If total number of runs scored in \( n \) matches is \( \left(\frac{\mathrm{n}+1}{4}\right)\left...

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