Explicație pas cu pas:
[tex]a = \frac{ \sqrt{1} - \sqrt{2}}{ \sqrt{1 \times 2} } + \frac{ \sqrt{2} - \sqrt{3}}{ \sqrt{2 \times 3} } + \frac{ \sqrt{3} - \sqrt{4}}{ \sqrt{3 \times 4} } + ... + \frac{ \sqrt{98} - \sqrt{99}}{ \sqrt{98 \times 99} }+ \frac{ \sqrt{99} - \sqrt{100}}{ \sqrt{99 \times 100} } = \\ [/tex]
[tex]= \frac{ \sqrt{1} }{ \sqrt{1} \times \sqrt{2}} - \frac{ \sqrt{2} }{ \sqrt{1} \times \sqrt{2} } + \frac{ \sqrt{2} }{ \sqrt{2} \times \sqrt{3}} - \frac{ \sqrt{3} }{ \sqrt{2} \times \sqrt{3} } + \frac{ \sqrt{3} }{ \sqrt{3} \times \sqrt{4}} - \frac{ \sqrt{4} }{ \sqrt{3} \times \sqrt{4} } + ... +\frac{ \sqrt{98} }{ \sqrt{98} \times \sqrt{99}} - \frac{ \sqrt{99} }{ \sqrt{98} \times \sqrt{99} } + \frac{ \sqrt{99} }{ \sqrt{99} \times \sqrt{100}} - \frac{ \sqrt{100} }{ \sqrt{99} \times \sqrt{100} } \\ [/tex]
[tex]= \frac{1}{ \sqrt{2} } - \frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{3} } - \frac{1}{ \sqrt{2} } + \frac{1}{ \sqrt{4} } - \frac{1}{ \sqrt{3} } + ... + \frac{1}{ \sqrt{99} } - \frac{1}{ \sqrt{98} } + \frac{1}{ \sqrt{100} } - \frac{1}{ \sqrt{99} } \\ [/tex]
[tex]= - \frac{1}{ \sqrt{1} } + \frac{1}{ \sqrt{100} } = \frac{1}{10} - 1 = - \frac{9}{10} \\ [/tex]
