Q. $\frac{1}{1!(n - 1)!} + \frac{1}{3!(n - 3)!} + \frac{1}{5!(n - 5)!} + ...$ is equal to
A) $\frac{2^{n - 4}}{n!}$ for even values of $n$ only
B) $\frac{2^{n - 4} + 1}{n!} - 1$ for odd values of $n$ only
C) $\frac{2^{n - 1}}{n!}$ for all values of $n$
D) None of the above
Explanation
(c)
Let$S = \frac{1}{n!}\left\lbrack \frac{n!}{1!(n - 1)!} + \frac{n!}{3!(n - 3)!} + \frac{n!}{5!(n - 5)!} + ... \right\rbrack$
$= \frac{1}{n!}\lbrack^{n}C_{1} +^{n}C_{3} +^{n}C_{5} + ...\rbrack$
$= \frac{1}{n!}2^{n - 1}for\ all\ value\ of\ n\ only$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.