Question

В конце 1/корень 80+корень 81​

Answer 1

Ответ:

Пошаговое объяснение:

$\frac{1}{1+\sqrt{2} }+ \frac{1}{\sqrt{2}+\sqrt{3} }+\frac{1}{\sqrt{3}+\sqrt{4} }+...+\frac{1}{\sqrt{80}+\sqrt{81} }=8$

$\frac{1}{1+\sqrt{2} }=\frac{1}{1+\sqrt{2} }(\frac{1-\sqrt{2}}{1-\sqrt{2}} )=\frac{1-\sqrt{2}}{1-2} =\sqrt2} -1;$$\frac{1}{\sqrt{2}+\sqrt{3} }=\frac{1}{\sqrt{2}+\sqrt{3} }(\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3} })=\frac{\sqrt{2}-\sqrt{3}}{2-3 }=\sqrt{3} -\sqrt{2} ;$$\frac{1}{\sqrt{3}+\sqrt{4} }=\frac{1}{\sqrt{3}+\sqrt{4} }(\frac{\sqrt{3}-\sqrt{4}}{\sqrt{3}-\sqrt{4} })=\frac{\sqrt{3}-\sqrt{4}}{3-4 }=\sqrt{4} -\sqrt{3}$

................................................................................

$\frac{1}{\sqrt{80}+\sqrt{81} }=\frac{1}{\sqrt{80}+\sqrt{81} }(\frac{\sqrt{80}-\sqrt{81}}{\sqrt{80}-\sqrt{81} })=\frac{\sqrt{80}-\sqrt{81}}{80-81 }=\sqrt{81} -\sqrt{80}$

∑

$\sqrt{81} -1=9-1=8$