Suma de Numere-cu fractii

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Matematica Ajutor :: Cereri si rezolvari de probleme :: Clasa a-VII-a

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$Suma de Numere-cu fractii Empty$ Suma de Numere-cu fractii

andy28 Dum Noi 18, 2012 9:55 pm

$S = \frac{1}{{3 \times 5}} + \frac{1}{{5 \times 7}} + \cdots + \frac{1}{{15 \times 17}}$

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$Suma de Numere-cu fractii Empty$ Re: Suma de Numere-cu fractii

Admin Dum Noi 18, 2012 10:10 pm

$\begin{array}{l} \frac{1}{{\left( {2k + 1} \right)\left( {2k + 3} \right)}} = \frac{1}{2}\left( {\frac{1}{{2k + 1}} - \frac{1}{{2k + 3}}} \right)\\ k = 1 \Rightarrow \frac{1}{{3 \times 5}} = \frac{1}{2}\left( {\frac{1}{3} - \frac{1}{5}} \right)\\ k = 2 \Rightarrow \frac{1}{{5 \times 7}} = \frac{1}{2}\left( {\frac{1}{5} - \frac{1}{7}} \right)\\ k = 3 \Rightarrow \frac{1}{{7 \times 9}} = \frac{1}{2}\left( {\frac{1}{7} - \frac{1}{9}} \right)\\ .........\\ k = 7 \Rightarrow \frac{1}{{15 \times 17}} = \frac{1}{2}\left( {\frac{1}{{15}} - \frac{1}{{17}}} \right)\\ S = \frac{1}{2}\left( {\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \cdots + \frac{1}{{15}} - \frac{1}{{17}}} \right) = \frac{1}{2}\left( {\frac{1}{3} - \frac{1}{{17}}} \right) = ... \end{array}$