Question

Rezolvarea completa va rog dau funda/coroana/inima

Answer 1

dupa reducerea  termenilor asemenea, raspunsul final este 1/√2-1/√20, care se poate rationaliza si scrie in diverse moduri
ai in pagina a doua descompunerea fiecarei fractii si apoi, insumate pe coloana,se vede mai clar

Answer 2

$\frac{ \sqrt{3} - \sqrt{2} }{\sqrt{6} } + \frac{ \sqrt{4} - \sqrt{3} }{\sqrt{12} } + \frac{ \sqrt{5} - \sqrt{4} }{\sqrt{20} } +...+ \frac{ \sqrt{20} - \sqrt{19} }{\sqrt{380} } = \\ \\ = \frac{ \sqrt{3} - \sqrt{2} }{\sqrt{3}\cdot \sqrt{2} } } + \frac{ \sqrt{4} - \sqrt{3} }{\sqrt{4}\cdot \sqrt{3} } + \frac{ \sqrt{5} - \sqrt{4} }{\sqrt{5}\cdot \sqrt{4} } +...+ \frac{ \sqrt{20} - \sqrt{19} }{\sqrt{20}\cdot \sqrt{19} } =$

$=\frac{\sqrt{3}} {\sqrt{3}\cdot \sqrt{2} } - \frac{\sqrt{2}} {\sqrt{3}\cdot \sqrt{2} } +\frac{\sqrt{4}} {\sqrt{4}\cdot \sqrt{3} } - \frac{\sqrt{3}} {\sqrt{4}\cdot \sqrt{3} } +\frac{\sqrt{5}} {\sqrt{5}\cdot \sqrt{4} } - \frac{\sqrt{4}} {\sqrt{5}\cdot \sqrt{4} }+...+ \\ \\ + \frac{\sqrt{20}} {\sqrt{20}\cdot \sqrt{19} } - \frac{\sqrt{19}} {\sqrt{20}\cdot \sqrt{19} }=$

$= \frac{1}{\sqrt{2} }- \frac{1}{\sqrt{3} }+\frac{1}{\sqrt{3} }-\frac{1}{\sqrt{4} }+ \frac{1}{\sqrt{4} }+\frac{1}{\sqrt{5} }-\frac{1}{\sqrt{5} }+...+\frac{1}{\sqrt{19} }-\frac{1}{\sqrt{20} }= \\ \\ = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{20}} = \frac{\sqrt{10}-1} {\sqrt{20}} = \frac{\sqrt{20}(\sqrt{10}-1)}{20} = \frac{2\sqrt{5}(\sqrt{10}-1)}{20} =\frac{\sqrt{5}(\sqrt{10}-1)}{10}$