Question

Rút gọn biểu thức: \(P=1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{n\left(n+1\right)}\)

giup nha, kb nhé mn mk có ít bn bè quá ko bít nói chuyện vs ai đây

Answer 1

\(P=1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{n\left(n+1\right)}\)

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

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

     \(=2-\frac{1}{n+1}=\frac{2\left(n+1\right)}{n+1}-\frac{1}{n+1}=\frac{2n+2-1}{n+1}=\frac{2n+1}{n+1}\)

Answer 2

\(P=1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+......+\frac{1}{n\left(n+1\right)}=1+1-\frac{1}{2}+\frac{1}{2}-.....-\frac{1}{\left(n+1\right)}\)

\(\Rightarrow P=2-\frac{1}{\left(n+1\right)}=\frac{2n+1}{n+1}\)