A function f(x) is given by f(x)

Question:

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

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

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

  3. (3) $\frac{39}{2}$

  4. (4) $\frac{29}{2}$


Correct Option: , 3

Solution:

$f(x)=\frac{5^{x}}{5^{x}+5} \ldots$ (i)

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

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

Adding equation (i) and(ii)

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

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

$f\left(\frac{2}{20}\right)+f\left(\frac{38}{20}\right)=1$

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

and $f\left(\frac{20}{20}\right)=f(1)=\frac{1}{2}$

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

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