$Iff:R\to Risgivenbyf\left(x\right)=\frac{9^x}{9^x+3},\forall x\in R,thenf\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)+\mathrm{.....}+f\left(\frac{1995}{1996}\right)=$

$\begin{array}{l}f\left(x\right)=\frac{9^x}{9^x+3},f(1-x)=\frac{9^{1-x}}{9^{1-x}+3}=\frac{9}{9+{3.9}^x}=\frac{3}{3+9^x}\\ \therefore f\left(x\right)+f(1-x)=\frac{9^x}{9^x+3}+\frac{3}{3+9^x}=1\\ \therefore f\left(\frac{1}{1997}\right)+f\left(\frac{2}{1997}\right)+---+f\left(\frac{1995}{1997}\right)+f\left(\frac{1996}{1997}\right)\\ =(f\left(\frac{1}{1997}\right)+f\left(\frac{1996}{1997}\right))+(f\left(\frac{2}{1997}\right)+f\left(\frac{1995}{1997}\right))+---\\ =1+1+---+998times\\ =998\end{array}$