Question

50 баллов за подробный ответ​

Answer 1

Ответ:

$\displaystyle 4$

Объяснение:

$\displaystyle \frac{1}{\sqrt 1+\sqrt 3}+\frac{1}{\sqrt 3+\sqrt 5}+\frac{1}{\sqrt 5+\sqrt 7}+...+\frac{1}{\sqrt {79}+\sqrt {81}}=\\\\\\\frac{\sqrt 1-\sqrt 3}{(\sqrt 1+\sqrt 3)(\sqrt 1-\sqrt 3)}+\frac{\sqrt 3-\sqrt 5}{(\sqrt 3+\sqrt 5)(\sqrt 3-\sqrt 5)}+\frac{\sqrt 5-\sqrt 7}{(\sqrt 5+\sqrt 7)(\sqrt 5-\sqrt 7)}+...+\frac{\sqrt {79}-\sqrt {81}}{(\sqrt {79}+\sqrt {81})(\sqrt {79}-\sqrt {81})}=$

$\displaystyle \frac{\sqrt 1-\sqrt 3}{1-3}+\frac{\sqrt 3-\sqrt 5}{3-5}+\frac{\sqrt 5-\sqrt 7}{5-7}+...+\frac{\sqrt {79}-\sqrt {81}}{79-81}=$

$\displaystyle \frac{\sqrt 1-\sqrt 3}{-2}+\frac{\sqrt 3-\sqrt 5}{-2}+\frac{\sqrt 5-\sqrt 7}{-2}+...+\frac{\sqrt {79}-\sqrt {81}}{-2}=\\\\\\-\frac{1}{2}\cdot \left(\sqrt 1-\sqrt 3+ \sqrt 3-\sqrt 5+\sqrt 5-\sqrt 7+...+\sqrt {79}-\sqrt {81}\right) =\\\\\\ -\frac{1}{2}\cdot \left(\sqrt 1-\sqrt {81}\right)= -\frac{1}{2}\cdot \left( 1-9\right)=-\frac{1}{2}\cdot \left( -8\right)=4$