\(A=1+\frac{1}{3}+\frac{1}{6}+...+\frac{1}{300}\)
\(=2\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{600}\right)\)
\(=2\left(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{24\times25}\right)\)
\(=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{24}-\frac{1}{25}\right)\)
\(=2\left(1-\frac{1}{25}\right)=2\times\frac{24}{25}=\frac{48}{25}\)