\(\text{ If }f(x)=\frac{{2}^{x}}{{2}^{x}+\sqrt{2}},x\in R\text{, then }\sum _{k=1}^{81}f\left(\frac{k}{82}\right)\text{ …

\(f(x)=\frac{2x}{{2}^{x}+\sqrt{2}}\\ f(x)+f(1-x)=1&f(1/2)=\frac{{2}^{1/2}}{{2}^{1/2}+\sqrt{2}}=1/2\\ \sum _{k=1}^{81}f\left(\frac{k}{82}\right)=f\left(\frac{1}{82}\right)+f\left(\frac{2}{82}\right)+f\left(\frac{3}{32}\right)+⋯f\left(\frac{81}{82}\right)\\ =40+f(\frac{1}{2})=40+\frac{1}{2}=\frac{81}{2}\)