A function f(x) is given by

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 :

  1. $\frac{19}{2}$

  2. $\frac{49}{2}$

  3. $\frac{29}{2}$

  4. $\frac{39}{2}$


Correct Option: , 4

Solution:

$f(x)=\frac{5^{x}}{5^{x}+5} \quad f(2-x)=\frac{5}{5^{x}+5}$

$f(x)+f(2-x)=1$

$\Rightarrow f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+\ldots+f\left(\frac{39}{20}\right)$

$=\left(f\left(\frac{1}{20}\right)+f\left(\frac{39}{20}\right)\right)+\ldots+\left(f\left(\frac{19}{20}\right)+f\left(\frac{21}{20}\right)+f\left(\frac{20}{20}\right)\right)$

$=19+\frac{1}{2}=\frac{39}{2}$

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