\(\Leftrightarrow\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{99}{100}\)
\(\Leftrightarrow1-\frac{1}{x+1}=\frac{99}{100}\)
\(\frac{100}{100}-\frac{1}{x+1}=\frac{99}{100}\)
\(\frac{1}{x+1}=\frac{1}{100}\)
\(\Rightarrow x+1=100\)
\(x=99\)