# $\frac{1}{1!(n-1)!} – \frac{1}{3!(n-3)!} + \frac{1}{5!(n-5)!} – …is \; equal \; to$

Q: $\large \frac{1}{1!(n-1)!} – \frac{1}{3!(n-3)!} + \frac{1}{5!(n-5)!} – …is \; equal \; to$

(A) $\large \frac{2^{n-1}}{n!}$ for all n ∈ N

(B) $\large \frac{2^{n-1}}{n!}$ for odd values of n only

(C) $\large \frac{2^{n-1}}{n!}$ for even values of n only

(D) none of these

Sol. Given expression $\large = \frac{1}{n!}(n_{C_1} – n_{C_3} + n_{C_5} -…..) = \frac{2^{n/2}sin\frac{n\pi}{4}}{n!}$