Question

puteti cu suma lui gaos ca nu mai pot o sa mor de nu explodez ​

Answer 1

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

putem scrie:

$\dfrac{\sqrt{3} - \sqrt{2} }{\sqrt{6} } = \dfrac{\sqrt{3} - \sqrt{2} }{\sqrt{3} \cdot \sqrt{2}} = \dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{3}}$

expresia devine:

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

$= \dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{3}} + \dfrac{1}{\sqrt{3}} - \dfrac{1}{\sqrt{4}} + \dfrac{1}{\sqrt{4}} - \dfrac{1}{\sqrt{5}} + ... + \dfrac{1}{\sqrt{19}} - \dfrac{1}{\sqrt{20}}$

se reduc termenii asemenea:

$= \dfrac{^{5\sqrt{2})} 1}{\sqrt{2}} - \dfrac{^{\sqrt{5})}1}{\sqrt{20}} = \dfrac{5\sqrt{2} }{5 \cdot 2} - \dfrac{\sqrt{5} }{\sqrt{100}} = \dfrac{5\sqrt{2} - \sqrt{5} }{10}$