\(\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{n}+\sqrt{n+1}}=10\)
\(\frac{\sqrt{4}-\sqrt{5}}{\left(\sqrt{4}+\sqrt{5}\right)\left(\sqrt{4}-\sqrt{5}\right)}+\frac{\sqrt{5}-\sqrt{6}}{\left(\sqrt{5}+\sqrt{6}\right)\left(\sqrt{5}-\sqrt{6}\right)}+...+\frac{\sqrt{n}-\sqrt{n+1}}{\left(\sqrt{n}+\sqrt{n+1}\right)\left(\sqrt{n}-\sqrt{n+1}\right)}=10\)
\(\frac{\sqrt{4}-\sqrt{5}}{4-5}+\frac{\sqrt{5}-\sqrt{6}}{5-6}+...+\frac{\sqrt{n}-\sqrt{n+1}}{n-\left(n+1\right)}=10\)
\(\frac{\sqrt{4}-\sqrt{5}}{-1}+\frac{\sqrt{5}-\sqrt{6}}{-1}+...+\frac{\sqrt{n}-\sqrt{n+1}}{-1}=10\)
\(\frac{\sqrt{4}-\sqrt{n+1}}{-1}=10\)
\(2-\sqrt{n+1}=-10\)
\(\sqrt{n+1}=12\)
\(\Rightarrow n+1=144\Rightarrow n=143\)
