Răspuns:
[tex][\frac{1}{\sqrt{2} +\sqrt{1} } +\frac{1}{\sqrt{3} +\sqrt{2} } +...+\frac{1}{\sqrt{n+1} +\sqrt{n} }] =2\\[/tex]
[tex][\frac{\sqrt{2} -\sqrt{1}}{(\sqrt{2} +\sqrt{1})(\sqrt{2} -\sqrt{1})} +\frac{\sqrt{3} -\sqrt{2}}{(\sqrt{3} +\sqrt{2})(\sqrt{3} -\sqrt{2})} +...+\frac{\sqrt{n+1} -\sqrt{n}}{(\sqrt{n+1} +\sqrt{n})(\sqrt{n+1} -\sqrt{n})} ] =2[/tex]
[tex][\frac{\sqrt{2} -\sqrt{1}}{2-1} +\frac{\sqrt{3} -\sqrt{2}}{3-2} +...+\frac{\sqrt{n+1} -\sqrt{n}}{n+1-n} ] =2[/tex]
[tex][\sqrt{2} -\sqrt{1} +\sqrt{3} -\sqrt{2} +\sqrt{4} -\sqrt{3} +...+\sqrt{n} -\sqrt{n-1} +\sqrt{n+1} -\sqrt{n} ]=2\\[/tex]
[tex][\sqrt{n+1} -1 ]=2\\[/tex]
⇔ [tex]2\leq \sqrt{n+1}-1 <3\\[/tex]
[tex]3\leq \sqrt{n+1} <4\\[/tex]
⇔ 9 ≤ n+1 < 16
⇔ 8 ≤ n < 15
n ∈{8, 9, 10, 11, 12, 13, 14}
