Prove that 1². C 0 + 2². C 1 + 3². C 2 + 4². C 3 +.... + (n+1)² C n = 2 n − 2 (n+1) (n+4).
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(1 + x) n = C 0 + C 1 x + C 2 x 2 + ........... + C n x n
multiply by x and then differentiate
(1 + x) n + x . n (1 + x) n – 1 = C 0 + 2.C 1 x + 3.C 2 x 2 + ........ + (n + 1) . C n x n
again multiply by x and then differentiate
(1 + x) n + nx (1 + x) n – 1 + 2nx (1 + x) n – 1 + n (n – 1) x 2 (1 + x) n – 2 = C 0 + 2 2 C 1 x + 3 2 C 2 x 2 +...+ (n+1) 2 C n x n
put x = 1
then S = 2 n + n. 2 n – 1 + 2n. 2 n – 1 + n (n – 1) 2 n – 2
= 2 n – 2 [4 + 2n + 4n + n 2 – n]
= 2 n – 2 (n + 1) (n + 4)
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