Réponse :
Bonsoir,
Explications étape par étape
[tex]\dfrac{1}{\sqrt{2} +\sqrt{1} } +\dfrac{1}{\sqrt{3} +\sqrt{2} } +\dfrac{1}{\sqrt{4} +\sqrt{3} } +...+\dfrac{1}{\sqrt{n+1} +\sqrt{n} } \\\\=\dfrac{\sqrt{2} -\sqrt{1}}{(\sqrt{2} +\sqrt{1})*(\sqrt{2} -\sqrt{1}) }+\dfrac{\sqrt{3} -\sqrt{2}}{(\sqrt{3} +\sqrt{2})*(\sqrt{3} -\sqrt{2}) }+....+\dfrac{\sqrt{n+1} -\sqrt{n}}{(\sqrt{n+1} +\sqrt{n})*(\sqrt{n+1} -\sqrt{n}) }\\\\=-\dfrac{\sqrt{1}} {1 }+\dfrac{\sqrt{n+1}} {1 }\\\\=\sqrt{n+1} -1\\\\\\\sqrt{n+1} -1 \geq 100\\\\\sqrt{n+1}\geq 101\\[/tex]
[tex]n+1 \geq 101^2\\\\n \geq 101^2 -1\\\\\boxed{n \geq 10200}[/tex]
