\(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{99\cdot101}\)
\(=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\)
\(=\dfrac{1}{3}-\dfrac{1}{101}=\dfrac{98}{303}\)
\(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+\dfrac{2}{7\cdot9}+...+\dfrac{2}{99\cdot101}\)
\(=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\)
\(=\dfrac{1}{3}-\dfrac{1}{101}\)
\(=\dfrac{101}{303}-\dfrac{3}{303}\)
\(=\dfrac{98}{303}\)
\(\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}+...+\dfrac{2}{99.101}\)
\(2A=2.\left(\dfrac{1}{3.5}+\dfrac{1}{5.7}+\dfrac{1}{7.9}+...+\dfrac{1}{99.101}\right)\)
\(2A=2.\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\)
\(2A=2.\left(\dfrac{1}{3}-\dfrac{1}{101}\right)\)
\(2A=2.\dfrac{98}{303}\)
\(2A=\dfrac{196}{303}\)
\(A=\dfrac{196}{303}:2\)
\(A=\dfrac{196}{303}.\dfrac{1}{2}\)
\(A=\dfrac{98}{303}\)
\(\dfrac{2}{3.5}\) + \(\dfrac{2}{5.7}\) + ..... + \(\dfrac{2}{99.101}\)
=\(\dfrac{1}{3}\) - \(\dfrac{1}{5}\) + \(\dfrac{1}{5}\) - \(\dfrac{1}{7}\) + ..... + \(\dfrac{1}{99}\) - \(\dfrac{1}{101}\)
=\(\dfrac{1}{3}\) - \(\dfrac{1}{101}\)
=\(\dfrac{98}{303}\)
