Question

2/x*(x+2)+2/3*5+2/5*7+...2/99*101=100/101

Answer 1

\(\frac{2}{x.\left(x+2\right)}+\frac{2}{3.5}+\frac{2}{5.7}.+.....+\frac{2}{99.101}=\frac{100}{101}\)

\(\Rightarrow\frac{1}{x\left(x+1\right)}+\frac{1}{3.5}+\frac{1}{5.7}+....+\frac{1}{99.101}=\frac{100}{101}\)

\(\Rightarrow\frac{1}{x}-\frac{1}{x+1}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+....+\frac{1}{99}-\frac{1}{101}=\frac{100}{101}\)

\(\Rightarrow\frac{1}{x}-\frac{1}{101}=\frac{100}{101}\)

\(\Rightarrow\frac{1}{x}=\frac{101}{101}\)

\(\Rightarrow\frac{1}{x}=\frac{1}{1}\)

\(\Rightarrow x=1\)