The value of frac{1}{1! 50!} + frac{1}{3! 48! | vedclass

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(B) The given expression is $S = \sum_{r=1}^{26} \frac{1}{(2r-1)! (51-(2r-1))!}$.Multiply and divide by $51!$:$S = \frac{1}{51!} \sum_{r=1}^{26} \frac

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$\frac{2^{50}}{51!}$

Vedclass · Answer

(B) The given expression is $S = \sum_{r=1}^{26} \frac{1}{(2r-1)! (51-(2r-1))!}$.Multiply and divide by $51!$:$S = \frac{1}{51!} \sum_{r=1}^{26} \frac{51!}{(2r-1)! (52-2r)!} = \frac{1}{51!} \sum_{r=1}^{26} {}^{51}C_{2r-1}$.This sum represents the sum of odd-indexed binomial coefficients of $(1+x)^{51}$:$S = \frac{1}{51!} ({}^{51}C_1 + {}^{51}C_3 + \dots + {}^{51}C_{51})$.We know that the sum of odd-indexed binomial coefficients is $2^{n-1}$. Here $n=51$,so the sum is $2^{51-1} = 2^{50}$.Therefore,$S = \frac{2^{50}}{51!}$.

The value of $\frac{1}{1! 50!} + \frac{1}{3! 48!} + \frac{1}{5! 46!} + \dots + \frac{1}{49! 2!} + \frac{1}{51! 1!}$ is $.............$.

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