If for a positive integer n , the sum _ r=1 ^n ( r+2 2^r ) ^n C _r = 7 3^n - 2^ n+1 2^n , then the value of n is equal to

Step-by-step solution

The given sum is _ r=1 ^n ( r+2 2^r ) ^n C _r . We can split this into two parts: _ r=1 ^n r 2^r ^n C _r + 2 _ r=1 ^n 1 2^r ^n C _r Using the identity r ^n C _r = n ^ n-1 C _ r-1 , the first sum becomes: _ r=1 ^n n 2^r ^ n-1 C _ r-1 = n 2 _ r=1 ^n 1 2^ r-1 ^ n-1 C _ r-1 = n 2 (1 + 1 2 )^ n-1 = n 2 ( 3 2 )^ n-1 The second sum is: 2 ( _ r=0 ^n 1 2^r ^n C _r - 1 ) = 2 ( (1 + 1 2 )^n - 1 ) = 2 ( 3 2 )^n - 2 Adding them together: n 2 ( 3 2 )^ n-1 + 2 ( 3 2 )^n - 2 = n 2 2 3 ( 3 2 )^n + 2 ( 3 2 )^n - 2 = ( n 3 + 2 ) ( 3