Question

Cho \(n\in N\)*. CMR

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

Answer 1

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

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

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