"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 :"

Question

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 = $19 + {1 \over 2} = {{39} \over 2}$"

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 :

Solution

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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 :

Solution

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