Question:
A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series
$\mathrm{f}\left(\frac{1}{20}\right)+\mathrm{f}\left(\frac{2}{20}\right)+\mathrm{f}\left(\frac{3}{20}\right)+\ldots \ldots+\mathrm{f}\left(\frac{39}{20}\right)$ is equal to :
Correct Option: , 4
Solution:
$f(x)=\frac{5^{x}}{5^{x}+5} \quad f(2-x)=\frac{5}{5^{x}+5}$
$f(x)+f(2-x)=1$
$\Rightarrow f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+\ldots+f\left(\frac{39}{20}\right)$
$=\left(f\left(\frac{1}{20}\right)+f\left(\frac{39}{20}\right)\right)+\ldots+\left(f\left(\frac{19}{20}\right)+f\left(\frac{21}{20}\right)+f\left(\frac{20}{20}\right)\right)$
$=19+\frac{1}{2}=\frac{39}{2}$