Explicație pas cu pas:
[tex]\frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{6} } + \frac{ \sqrt{4} - \sqrt{3} }{ \sqrt{12} } + ... + \frac{ \sqrt{10} - \sqrt{9} }{ \sqrt{90} } = \\[/tex]
[tex]= \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} \cdot \sqrt{2} } + \frac{ \sqrt{4} - \sqrt{3} }{ \sqrt{4} \cdot \sqrt{3} } + ... + \frac{ \sqrt{10} - \sqrt{9} }{ \sqrt{10} \cdot \sqrt{9} }\\[/tex]
[tex]= \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{10}}{ \sqrt{10} \cdot \sqrt{9} } - \frac{\sqrt{9} }{ \sqrt{10} \cdot \sqrt{9} }\\[/tex]
[tex]= \frac{1}{\sqrt{2} } - \frac{1}{ \sqrt{3}} + \frac{1}{\sqrt{3} } - \frac{1}{ \sqrt{4}} + ... + \frac{1}{\sqrt{9}} - \frac{1}{ \sqrt{10}}\\[/tex]
[tex]= \frac{1}{ \sqrt{2} } - \frac{1}{\sqrt{10}} = \frac{ \sqrt{5} }{ \sqrt{10} } - \frac{1}{ \sqrt{10} } = \bf \frac{ \sqrt{5} - 1}{ \sqrt{10} } \\ [/tex]
q.e.d.
