A cricket player played n ( n 1 ) matches during his career and made a total of 4 ( n + 1 ) ( 2 n + 1 − n − 2 ) runs. If the player made k . 2 n − k + 1 runs in the k th match ( 1 ≤ k ≤ n ) . Then

Question

A cricket player played n ( n > 1 ) matches during his career and made a total of 4 ( n + 1 ) ( 2 n + 1 − n − 2 ) ​ runs. If the player made k . 2 n − k + 1 runs in the k th match ( 1 ≤ k ≤ n ) . Then the value of ' n ' is

TestHub · Accepted Answer

Where S = 1. 2 n + 2. 2 n − 1 + 3. 2 n − 2 + … . + n .2 ​ 2 1 ​ S = 2 n − 1 + 2. 2 n − 2 + … .. + ( n − 1 ) .2 + n ​ ​ S u b t r a c t in g , 2 1 ​ S = ( 2 n + 2 n − 1 + … .. + 2 ) − n = 2 ( 2 n − 1 ) − n ∴ S = 2 ( 2 n + 1 − 2 ) − 2 n = 2 ( 2 n − 1 − n − 2 ) Hence, 2 n − 1 ​ = 2 ( i.e., ) n = 7

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