QJEEM21Maeda9: JEE Main 2021 Mathematics - Class 11 JEEM21Maeda9 Solution: A function f(x) is given by , then the sum of the series is…

Question

Verified Solution • 2025–26 Session moderate

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

\begin{aligned} : \\ : \end{aligned}

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}

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