Question

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

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

Answer 1

Answer: (D)

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