If C r stands for n C r , then the sum of the series 2 ( n 2 )! ( n 2 )! n ! [ C _ 0 ^ 2…
Mathematics · JEE Advanced · NTA Exams — Binomial Theorem And Its Simple Applications
If Cr stands for nCr, then the sum of the series
\[\frac{2\left(\frac{\mathrm{n}}{2}\right)!\left(\frac{\mathrm{n}}{2}\right)!}{\mathrm{n}!}\left[\mathrm{C}_{0}^{2}-2 \mathrm{C}_{1}^{2}+3 \mathrm{C}_{2}^{2}-\ldots+(-1)^{\mathrm{n}}(\mathrm{n}+1) \mathrm{C}_{\mathrm{n}}^{2}\right]\]
where n is an even positive integer, is equal to :
\[\frac{2\left(\frac{\mathrm{n}}{2}\right)!\left(\frac{\mathrm{n}}{2}\right)!}{\mathrm{n}!}\left[\mathrm{C}_{0}^{2}-2 \mathrm{C}_{1}^{2}+3 \mathrm{C}_{2}^{2}-\ldots+(-1)^{\mathrm{n}}(\mathrm{n}+1) \mathrm{C}_{\mathrm{n}}^{2}\right]\]
where n is an even positive integer, is equal to :
- (–1)n/2 (n + 2)
- (–1)n (n + 1)
- (–1)n/2 (n + 1)
- none of these
Answer
(A) (–1) n/2 (n + 2)
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