Question

Va rog dau multe puncte!
Cu rezolvare completa si expicata.​

Answer 1

Explicație pas cu pas:

$\dfrac{ \sqrt{2} - 1}{\sqrt{2} } + \dfrac{ \sqrt{3} - \sqrt{2} }{\sqrt{6} } + \dfrac{ \sqrt{4} - \sqrt{3} }{\sqrt{12} } + ... + \dfrac{ \sqrt{8} - \sqrt{7} }{\sqrt{56} } =$

$= \dfrac{ \sqrt{2} - 1}{\sqrt{2 \cdot 1} } + \dfrac{ \sqrt{3} - \sqrt{2} }{\sqrt{3 \cdot 2} } + \dfrac{ \sqrt{4} - \sqrt{3} }{\sqrt{4 \cdot 3} } + ... + \dfrac{ \sqrt{8} - \sqrt{7} }{\sqrt{8 \cdot 7} }$

$= \dfrac{ \sqrt{2} - 1}{\sqrt{2} \cdot \sqrt{1} } + \dfrac{ \sqrt{3} - \sqrt{2} }{\sqrt{3} \cdot \sqrt{2} } + \dfrac{ \sqrt{4} - \sqrt{3} }{\sqrt{4} \cdot \sqrt{3} } + ... + \dfrac{ \sqrt{8} - \sqrt{7} }{\sqrt{8} \cdot \sqrt{7} }$

$= \dfrac{ \sqrt{2}}{\sqrt{2} \cdot \sqrt{1} } - \dfrac{1}{\sqrt{2} \cdot \sqrt{1} } + \dfrac{ \sqrt{3}}{\sqrt{3} \cdot \sqrt{2} } - \dfrac{\sqrt{2} }{\sqrt{3} \cdot \sqrt{2} } + \dfrac{ \sqrt{4}}{\sqrt{4} \cdot \sqrt{3} } - \dfrac{\sqrt{3} }{\sqrt{4} \cdot \sqrt{3} } + ... + \dfrac{ \sqrt{8}}{\sqrt{8} \cdot \sqrt{7} } - \dfrac{\sqrt{7} }{\sqrt{8} \cdot \sqrt{7} }$

$= \dfrac{1}{\sqrt{1} } - \dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2} } - \dfrac{1}{\sqrt{3}} + \dfrac{1}{\sqrt{3} } - \dfrac{1}{\sqrt{4}} + ... + \dfrac{1}{\sqrt{7} } - \dfrac{1}{\sqrt{8}}$

$= 1 - \dfrac{1}{\sqrt{8}} \bf < 1$

q.e.d.