Correct option is (1) \(\frac{81}{2}\)
\(f(x)=\frac{2^x}{2^x+2^{1 / 2}}=\frac{2^x}{2^x+\sqrt{2}}\)
\(f(1-x)=\frac{2^{1-x}}{2^{1-x}+2^{1 / 2}}=\frac{\frac{2}{2^x}}{\frac{2}{2^x}+2^{1 / 2}}=\frac{2}{2+\sqrt{2} 2^x} \)
\(=\frac{\sqrt{2}}{2^x+\sqrt{2}} \)
\(\Rightarrow f(x)+f(1-x)=\frac{\sqrt{2}+2^x}{\sqrt{2}+2^x}=1\)
\( \Rightarrow \sum_{k=1}^{81} f\left(\frac{k}{82}\right)+f\left(\frac{2}{82}\right)+\left(f\left(\frac{3}{82}\right)\right)+\ldots \ldots+f\left(\frac{40}{82}\right)+f\left(\frac{41}{82}\right)+f\left(\frac{42}{82}\right) +\ldots+f\left(\frac{79}{82}\right)+f\left(\frac{80}{82}\right)+f\left(\frac{81}{82}\right) \)
\( =\left[f\left(\frac{1}{82}\right)+f\left(\frac{81}{82}\right)\right]+\left[f\left(\frac{2}{82}\right)+f\left(\frac{80}{82}\right)\right]+\ldots +\left[f\left(\frac{40}{82}\right)+f\left(\frac{42}{82}\right)+f\left(\frac{41}{82}\right)\right] \)
