Question

Chứng minh : \(\frac{1}{n\left(n+1\right)}\)= \(\frac{1}{n}-\frac{1}{n+1}\)

Answer 1

\(\frac{1}{n}-\frac{1}{n+1}\)

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

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

\(\Rightarrow DPCM\)

Answer 2

Ta có:  \(\frac{1}{n}-\frac{1}{n+1}=\frac{n+1-n}{n\left(n+1\right)}=\frac{\left(n-n\right)+1}{n\left(n+1\right)}=\frac{1}{n\left(n+1\right)}\)

Vì \(\frac{1}{n\left(n+1\right)}=\frac{1}{n\left(n+1\right)}\Rightarrow\frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1}\) ĐPCM

Answer 3

Ta có:

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

Answer 4

Ta có:\(\frac{1}{n\left(n+1\right)}=\frac{\left(n+1\right)-n}{n\left(n+1\right)}=\frac{n+1}{n\left(n+1\right)}-\frac{n}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1}\)

Vậy ta có điều phải chứng minh: \(\frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1}\)