Question

A function $f ( x )$ is given by $f ( x )=\frac{5^{ x }}{5^{ x }+5},$
then the sum of the series
$f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots +f\left(\frac{39}{20}\right)$ is equal to:

• A. $\frac{19}{2}$
• B. $\frac{49}{2}$
• C. $\frac{29}{2}$
• D. $\frac{39}{2}$

Answer 1

Answer: (D)

$f(x)=\frac{5^{x}}{5^{x}+5} $
$ f(2-x)=\frac{5}{5^{x}+5}$
$f(x)+f(2-x)=1$
$\Rightarrow f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+\ldots+f\left(\frac{39}{20}\right)$
$=\left(f\left(\frac{1}{20}\right)+f\left(\frac{39}{20}\right)\right)+\ldots+\left(f\left(\frac{19}{20}\right)+f\left(\frac{21}{20}\right)+f\left(\frac{20}{20}\right)\right)$
$=19+\frac{1}{2}=\frac{39}{2}$