The sum S = (1/9!) + (1/3!7!) + (1/5!5!) + (1/7!3!) + (1/9!) is equal to

Question

The sum S = (1/9!) + (1/3!7!) + (1/5!5!) + (1/7!3!) + (1/9!) is equal to (A)  (210 / 8!) (B) (29 /10!) (C) (27 /10!) (D) (26 / 10!)

Tardigrade Admin · Accepted Answer

Let S =(1/9 !)+(1/3 ! 7 !)+(1/5 ! 5 !)+(1/7 ! 3 !)+(1/9 !) =(1/10 !)[(10 !/9 !)+(10 !/3 ! 7 !)+(10 !/5 ! 5 !)+(10 !/7 ! 3 !)+(10 !/9 !)] =(1/10 !)[ 10 C1+ 10 C3+ 10 C5+ 10 C7+ 10 C9] = (1/10 !)(210-1) [∵ C1+C3+C5+ ldots=2n-1] = (29/10 !)

Q. The sum $S = \frac{1}{9 !} + \frac{1}{3 ! 7 !} + \frac{1}{5 ! 5 !} + \frac{1}{7 ! 3 !} + \frac{1}{9 !}$ is equal to

$\frac{2 ^{10}}{8 !}$

$\frac{2 ^{9}}{10 !}$

$\frac{2 ^{7}}{10 !}$

$\frac{2 ^{6}}{10 !}$

$\frac{2 ^{5}}{8 !}$

Q. The sum S=9!1​+3!7!1​+5!5!1​+7!3!1​+9!1​ is equal to

8!210​

10!29​

10!27​

10!26​

8!25​