A function f(x) is given by f(x)=frac{5^{x}}{5^{x}+5} , then sum of series fleft(frac{1}{20}right)+f

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1.Relation and Function

medium

A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series

$f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right)$ is equal to ....... .

$\frac{19}{2}$

$\frac{49}{2}$

$\frac{29}{2}$

$\frac{39}{2}$

(JEE MAIN-2021)

Solution

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

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

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

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

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

Standard 12

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(JEE MAIN-2014)

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