1.

If `C_r` stands for `nC_r`, then the sum of the series `(2(n/2)!(n/2)!)/(n !)[C_0^2-2C_1^2+3C_2^2-........+(-1)^n(n+1)C_n^2]` ,where n is an even positive integer, isA. `(-1)^(n//2) (n + 2)`B. `(-1)^(n) (n + 1)`C. `(-1)^(n//2) (n + 1)`D. None of these

Answer» We have, `C_(0)^(2)-2C_(1)^(2)+3C_(2)^(2)-4C_(3)^(2)+...+(-1)^(n)(n+1)C_(n)^(2)`
`=[C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+...+(-1)^(n)C_(n)^(2)]+[C_(1)^(2)-2C_(2)^(2)+3C_(3)^(2)-...+(-1)^(n)nC_(n)^(2)]`
`=(-1)^(n//2)(n!)/(((n)/(2))!((n)/(2))!)-(-1)^((n)/(2)-1)(n)/(2)(n!)/(((n)/(2))!((n)/(2))!)`
`=(-1)^(n//2)(n!)/(((n)/(2))!((n)/(2))!)(1+(n)/(2))`
`:.(2((n)/(2))!((n)/(2))!)/(n!)[C_(0)^(2)-2C_(1)^(2)+3C_(2)^(2)-...+(-1)^(r)(n+1)C_(n)^(2)]`
`=(2((n)/(2))!((n)/(2))!)/(n!)(-1)^(n//2)(n!)/(((n)/(2))!((n)/(2))!)((n+2))/(2)=(-1)^(n//2)(n+2)`


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