[tex] A = \dfrac{1}{\sqrt1+\sqrt2} + \dfrac{1}{\sqrt2+\sqrt3}+...+\dfrac{1}{\sqrt{99}+\sqrt{100}} \\
\text{Rationalizam si o sa avem} \\ \\
A = \dfrac{\sqrt1-\sqrt2}{1-2} + \dfrac{\sqrt2-\sqrt3}{2-3}+...+\dfrac{\sqrt{99}-\sqrt{100}}{99-100}}\\ \\
A = \dfrac{\sqrt1-\sqrt2}{-1} + \dfrac{\sqrt2-\sqrt3}{-1}+...+\dfrac{\sqrt{99}-\sqrt{100}}{-1}} \\ \\ \\
A = -\sqrt1+\sqrt2 -\sqrt2+\sqrt3-....-\sqrt{99}+\sqrt{100} \\ \\
\text{O sa se reduca teremnii si o sa ramana } \to \sqrt1~\text{si}~\sqrt{100} \\ \\
[/tex][tex] \downarrow \\ \\ \boxed{A = -\sqrt1+\sqrt{100}} \\ \\
\boxed{A = -1 + 10} \\ \\
\boxed{A = 9 \in \mathbb{N}} [/tex]