Question

Вычислить 1/корень1+корень2 +1/корень2+корен.3+...+1/корень99+корень100

Answer 1

$\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots \frac{1}{\sqrt{99} + \sqrt{100}} = ?\\\frac{1}{\sqrt{n} + \sqrt{n + 1}} = \frac{\sqrt{n + 1} - \sqrt{n}}{(\sqrt{n} + \sqrt{n + 1})(\sqrt{n + 1} - \sqrt{n})} = \sqrt{n + 1} - \sqrt{n}\\\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \cdots \frac{1}{\sqrt{99} + \sqrt{100}} = \sqrt{2} - \sqrt{1} + \sqrt{3} - \sqrt{2} + \cdots + \sqrt{100} - \sqrt{99} = \sqrt{100} - \sqrt{1} = 10 - 1 = 9$