\(\dfrac{1}{5}+\dfrac{1}{50}+\dfrac{1}{150}+...+\dfrac{1}{9500}\)
\(=\dfrac{1}{5}+\dfrac{1}{25}\left(\dfrac{1}{2}+\dfrac{1}{6}+...+\dfrac{1}{380}\right)\)
\(=\dfrac{1}{5}+\dfrac{1}{25}\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{19.20}\right)\)
\(=\dfrac{1}{5}+\dfrac{1}{25}\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{19}-\dfrac{1}{20}\right)\)
\(=\dfrac{1}{5}+\dfrac{1}{25}\left(1-\dfrac{1}{20}\right)=\dfrac{1}{5}+\dfrac{19}{500}=\dfrac{119}{500}\)
\(=\dfrac{1}{5}+\dfrac{1}{5\times10}+\dfrac{1}{10\times15}+...+\dfrac{1}{95\times100}\\ =\dfrac{1}{5}+\dfrac{1}{5}\times\left(\dfrac{5}{5\times10}+\dfrac{5}{10\times15}+...+\dfrac{5}{95\times100}\right)\\ =\dfrac{1}{5}+\dfrac{1}{5}\times\left(\dfrac{1}{5}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{15}+...+\dfrac{1}{95}-\dfrac{1}{100}\right)\\ =\dfrac{1}{5}+\dfrac{1}{5}\times\left(\dfrac{1}{5}-\dfrac{1}{100}\right)=\dfrac{1}{5}+\dfrac{1}{5}\times\dfrac{19}{100}\\ =\dfrac{1}{5}+\dfrac{19}{500}=\dfrac{119}{500}\)
