\(A=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^8}\)
=>3A=\(1+\frac{1}{3}+...+\frac{1}{3^7}\)
=>3A-A=\(\left(1+\frac{1}{3}+...+\frac{1}{3^7}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^8}\right)\)
=>2A=\(1-\frac{1}{3^8}=\frac{3^8-1}{3^8}\)
=>A=\(\frac{3^8-1}{3^8}:2=\frac{3^8-1}{2.3^8}\)