Question

S=1/1.2+1/2.3+1/3.4+...+1/99.100

 

Answer 1

S=1/1.2+1/2.3+1/3.4+...+1/99.10 

\(S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)

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

\(\Rightarrow S=\frac{1}{1}-\frac{1}{100}=\frac{99}{100}\)

Answer 2

\(S=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{99\cdot100}\)

\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)

\(S=1-\frac{1}{100}\)

\(S=\frac{99}{100}\)