Question

2/3.5+2/5.7+...+2/99.101

Answer 1

\(=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}=\dfrac{1}{3}-\dfrac{1}{101}=\dfrac{98}{303}\)

Answer 2

\(\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\)

\(\dfrac{2}{3.5}=\dfrac{1}{3}-\dfrac{1}{5}\)

\(\dfrac{2}{5.7}=\dfrac{1}{5}-\dfrac{1}{7}\)

\(\dfrac{2}{99.101}=\dfrac{1}{99}-\dfrac{1}{101}\)

\(\Rightarrow=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\)

\(=\dfrac{1}{3}-\dfrac{1}{101}\)

\(=\dfrac{98}{303}\)

Answer 3

\(\dfrac{2}{3.5} + \dfrac{2}{5.7} + ..... + \dfrac{2}{99.101}\) 

\(= \dfrac{1}{3} - \dfrac{1}{5} + \dfrac{1}{5} - \dfrac{1}{7} + ........ + \dfrac{1}{99} - \dfrac{1}{101}\)

\(= \dfrac{1}{3} - ( \dfrac{1}{5} - \dfrac{1}{5} ) + ........ + ( \dfrac{1}{99} -\dfrac{1}{99} ) - \dfrac{1}{101} \) 

\(= \dfrac{1}{3} - \dfrac{1}{101}\)

\(= \dfrac{101}{303} - \dfrac{3}{303}\)

\(= \dfrac{98}{303}\)