[tex]\frac{1}{\sqrt n+\sqrt{n+1}}=\frac{1}{\sqrt n+\sqrt{n+1}}\cdot\frac{\sqrt n-\sqrt{n+1}}{\sqrt n-\sqrt{n+1}}=[/tex]
[tex]\frac{\sqrt n-\sqrt{n+1}}{n-(n+1)}=\frac{\sqrt n-\sqrt{n+1}}{n-n-1}=\frac{\sqrt n-\sqrt{n+1}}{-1}=\sqrt{n+1}-\sqrt n[/tex]
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[tex]\frac{1}{\sqrt1+\sqrt2}+\frac{1}{\sqrt2+\sqrt3}+\frac{1}{\sqrt3+\sqrt4}+...+\frac{1}{\sqrt{24}+\sqrt{25}}=[/tex]
[tex]\sqrt2-\sqrt1+\sqrt3-\sqrt2+\sqrt4-\sqrt3+...+\sqrt{24}-\sqrt{23}+\sqrt{25}-\sqrt{24}=\\\\-\sqrt1+\sqrt{25}=-1+5=4=2^2[/tex]
