Question

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

• A. $2^{n}$
• B. $2^{n} /(n-1) !$
• C. $2^{n} / n !$
• D. $2^{n-1} / n !$

Answer 1

Answer: (D)

$S=\frac{1}{1 !(n-1) !}+\frac{1}{3 !(n-3) !}+\frac{1}{5 !(n-5) !}+\cdots$
Multiplying each term by $n ! / n !, S$ reduces to
$S=\frac{1}{n !}\left[\frac{n !}{1 !(n-1) !}+\frac{1}{3 !} \frac{n !}{(n-3) !}+\frac{1}{5 !} \frac{n !}{(n-5) !}+\cdots\right]$
$=\frac{1}{n !}\left[{ }^{n} C_{1}+{ }^{n} C_{3}+{ }^{n} C_{5}+\cdots\right]$
$=\frac{2^{n-1}}{n !}$