[tex]\it \dfrac{^{\sqrt{n+1}-\sqrt n)}1}{\sqrt{n(n+1)}(\sqrt{n+1}+\sqrt n)} =\dfrac{\sqrt{n+1}-\sqrt n}{\sqrt n\sqrt{n+1}}=\dfrac{1}{\sqrt n}-\dfrac{1}{\sqrt{n+1}}\ \ \ \ \ (*)[/tex]
[tex]\it (*)\ \Rightarrow a=\dfrac{1}{\sqrt1} -\dfrac{1}{\sqrt2}+\dfrac{1}{\sqrt2}-\dfrac{1}{\sqrt3}+\ ...\ +\dfrac{1}{\sqrt n}-\dfrac{1}{\sqrt{n+1}}=\\ \\ \\ =1-\dfrac{1}{\sqrt{n+1}} =1-\dfrac{\sqrt{n+1}}{n+1} =\dfrac{n+1-\sqrt{n+1}}{n+1}[/tex]
