frac{1}{1!(n - 1)!} + frac{1}{3!(n - 3)!} + f | vedclass

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Question

(B) Multiply the entire expression by $\frac{n!}{n!}$:$\frac{1}{n!} \left[ \frac{n!}{1!(n - 1)!} + \frac{n!}{3!(n - 3)!} + \frac{n!}{5!(n - 5)!} + \do

Vedclass · Accepted Answer

$\frac{2^{n - 1}}{n!}$; for all values of $n$

Vedclass · Answer

(B) Multiply the entire expression by $\frac{n!}{n!}$:$\frac{1}{n!} \left[ \frac{n!}{1!(n - 1)!} + \frac{n!}{3!(n - 3)!} + \frac{n!}{5!(n - 5)!} + \dots \right]$This simplifies to:$\frac{1}{n!} \left[ ^nC_1 + ^nC_3 + ^nC_5 + \dots \right]$We know that the sum of odd-indexed binomial coefficients is $^nC_1 + ^nC_3 + ^nC_5 + \dots = 2^{n - 1}$.Therefore,the expression equals $\frac{2^{n - 1}}{n!}$.

$\frac{1}{1!(n - 1)!} + \frac{1}{3!(n - 3)!} + \frac{1}{5!(n - 5)!} + \dots = $

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