A function f(x) is given by [f(x) = 5^x 5^x + 5], then the sum of the series [f( 1 20 ) + …

Question

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

Accepted Answer

$$f(x) = {{{5^x}} \over {{5^x} + 5}}$$ ..... (i)

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

$$f(2 - x) = {5 \over {{5^x} + 5}}$$ .... (ii)

Adding equation (i) and (ii)

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

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

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

$$\eqalign{
  \& :  \cr
  \& : \cr} $$

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

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

$$ \therefore $$ Sum = $$19 + {1 \over 2} = {{39} \over 2}$$

Solutions · 1