Explicație:
[tex]\frac{1}{1\cdot 2} + \frac{1}{2 \cdot 3} + ... + \frac{1}{23 \cdot 24} + \frac{1}{24 \cdot 25} = \\[/tex]
[tex]= \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + ... + \frac{1}{23} - \frac{1}{24} + \frac{1}{24} - \frac{1}{25} \\[/tex]
[tex]= \frac{1}{1} - \frac{1}{25} = \frac{25 - 1}{25} = \frac{24}{25} \\ [/tex]
[tex]\frac{1}{25 \cdot 26} + \frac{1}{26 \cdot 27} + ... + \frac{1}{48 \cdot 49} + \frac{1}{49 \cdot 50} = \\[/tex]
[tex]= \frac{1}{25} - \frac{1}{26} + \frac{1}{26} - \frac{1}{27} + ... + \frac{1}{48} - \frac{1}{49} + \frac{1}{49} - \frac{1}{50} \\[/tex]
[tex]= \frac{1}{25} - \frac{1}{50} = \frac{2 - 1}{50} = \frac{1}{50} \\ [/tex]
=>
[tex]x = \sqrt{ \frac{1}{94} \left( \frac{24}{25} - \frac{1}{50} \right)} = \sqrt{ \frac{1}{94} \cdot \frac{48 - 1}{50}} = \\ [/tex]
[tex]= \sqrt{ \frac{1}{94} \cdot \frac{47}{50}} = \sqrt{ \frac{1}{100} } = \bf \frac{1}{10} \\ [/tex]
q.e.d.
