Упростите выражение:
$\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \frac{1}{\sqrt{5} + \sqrt{4}} + ... + \frac{1}{\sqrt{100} + \sqrt{99}}$.
$\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \frac{1}{\sqrt{5} + \sqrt{4}} + ... + \frac{1}{\sqrt{100} + \sqrt{99}} = \frac{1(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} + \frac{1(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} + \frac{1(\sqrt{4} - \sqrt{3})}{(\sqrt{4} + \sqrt{3})(\sqrt{4} - \sqrt{3})} + \frac{1(\sqrt{5} - \sqrt{4})}{(\sqrt{5} + \sqrt{4})(\sqrt{5} - \sqrt{4})} + ... + \frac{1(\sqrt{100} - \sqrt{99})}{(\sqrt{100} + \sqrt{99})(\sqrt{100} - \sqrt{99})} = \frac{\sqrt{2} - 1}{2 - 1} + \frac{\sqrt{3} - \sqrt{2}}{3 - 2} + \frac{\sqrt{4} - \sqrt{3}}{4 - 3} + \frac{\sqrt{5} - \sqrt{4}}{5 - 4} + ... + \frac{\sqrt{100} - \sqrt{99}}{100 - 99} = \sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} + \sqrt{5} - \sqrt{4} + ... + \sqrt{100} - \sqrt{99} = -1 + 10 = 9$
Пожауйста, оцените решение
