$\begin{aligned} & f(\mathrm{x})=\frac{5^{\mathrm{x}}}{5^{\mathrm{x}}+5} \\ & f(2-\mathrm{x})=\frac{5}{5^{\mathrm{x}}+5} \\ & f(\mathrm{x})+f(2-\mathrm{x})=1 \\ & \Rightarrow \quad f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+\ldots .+f\left(\frac{39}{20}\right) \\ & =\left(f\left(\frac{1}{20}\right)+f\left(\frac{39}{20}\right)\right)+\ldots .+\left(f\left(\frac{19}{20}\right)+f\left(\frac{21}{20}\right)+f\left(\frac{20}{20}\right)\right) \\ & =19+\frac{1}{2}=\frac{39}{2}\end{aligned}$