If _ r=1 ^9 r^2 + 2 2^r ^9 C _r = ( 3 2 )^9 - , where , N , then the value of + is equal to

Step-by-step solution

The given sum is _ r=1 ^9 r^2 + 2 2^r ^9 C _r . We can rewrite the numerator as r^2 + 2 = r(r-1) + r + 2 . Splitting the sum into three parts: _ r=2 ^9 r(r-1) 2^r ^9 C _r + _ r=1 ^9 r 2^r ^9 C _r + 2 _ r=1 ^9 1 2^r ^9 C _r Using the identities r(r-1) ^n C _r = n(n-1) ^ n-2 C _ r-2 and r ^n C _r = n ^ n-1 C _ r-1 , we get: First sum: _ r=2 ^9 9 8 2^r ^7 C _ r-2 = 72 4 _ r=2 ^9 1 2^ r-2 ^7 C _ r-2 = 18 (1 + 1 2 )^7 = 18 ( 3 2 )^7 Second sum: _ r=1 ^9 9 2^r ^8 C _ r-1 = 9 2 _ r=1 ^9 1 2^ r-1 ^8 C _ r-1 = 9 2 (1 + 1 2