\(A=\dfrac{2}{3\text{x}6}+\dfrac{2}{6\text{x}9}+...+\dfrac{2}{99\text{x}102}\)
\(=\dfrac{2}{3}\text{x}\left(\dfrac{3}{3\text{x}6}+\dfrac{3}{6\text{x}9}+...+\dfrac{3}{99\text{x}102}\right)\)
\(=\dfrac{2}{3}\text{x}\left(\dfrac{1}{3}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{9}+...+\dfrac{1}{99}-\dfrac{1}{102}\right)\)
\(=\dfrac{2}{3}\text{x}\left(\dfrac{1}{3}-\dfrac{1}{102}\right)=\dfrac{2}{3}\text{x}\dfrac{33}{102}=\dfrac{2}{102}\text{x}\dfrac{33}{3}=11\text{x}\dfrac{1}{51}=\dfrac{11}{51}\)
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