Question

Answer 1

\(\Rightarrow1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}-2x=\dfrac{1}{2}\\ \Rightarrow1-\dfrac{1}{100}-\dfrac{1}{2}=2x\\ \Rightarrow2x=\dfrac{49}{100}\Rightarrow x=\dfrac{49}{200}\)

Answer 2

\(\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{99.100}\right)-2x=\dfrac{1}{2}\)

\(\Rightarrow\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)-2x=\dfrac{1}{2}\)

\(\Rightarrow2x=1-\dfrac{1}{100}-\dfrac{1}{2}\)

\(\Rightarrow2x=\dfrac{49}{100}\Rightarrow x=\dfrac{49}{200}\)