Răspuns:
n = 12
Explicație pas cu pas:
folosim formula:
[tex]\frac{1}{2n - 1} - \frac{1}{2n + 1} = \frac{2n + 1 - (2n - 1)}{(2n - 1)(2n + 1)} = \frac{2}{(2n - 1)(2n + 1)} \\ [/tex]
[tex] \frac{1}{2n - 1} - \frac{1}{2n + 1} = 2 \cdot \frac{1}{(2n - 1)(2n + 1)} \\ [/tex]
[tex]\boxed { \red{\frac{1}{(2n - 1)(2n + 1)} = \frac{1}{2} \cdot \Big(\frac{1}{2n - 1} - \frac{1}{2n + 1}\Big)}} \\ [/tex]
→
[tex]\frac{1}{3} + \frac{1}{15} + \frac{1}{35} + ... + \frac{1}{(2n - 1)(2n + 1)} = \frac{12}{25} \\ [/tex]
[tex]\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... + \frac{1}{(2n - 1)(2n + 1)} = \frac{12}{25} \\ [/tex]
[tex]\frac{1}{2} \cdot \Big(\frac{1}{1} - \frac{1}{3}\Big) + \frac{1}{2} \cdot \Big(\frac{1}{3} - \frac{1}{5}\Big) + \frac{1}{2} \cdot \Big(\frac{1}{5} - \frac{1}{7}\Big) + ... + \frac{1}{2} \cdot \Big(\frac{1}{2n - 1} - \frac{1}{2n + 1}\Big) = \frac{12}{25} \\ [/tex]
[tex]\frac{1}{2} \cdot \Big(\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + ... + \frac{1}{2n - 1} - \frac{1}{2n + 1}\Big) = \frac{12}{25} \\ [/tex]
[tex]\frac{1}{1} - \frac{1}{2n + 1} = \frac{24}{25} \iff \frac{2n + 1 - 1}{2n + 1} = \frac{24}{25}\\ [/tex]
[tex]\frac{2n}{2n + 1} = \frac{24}{24 + 1} \iff 2n = 24\\ [/tex]
[tex]\implies \red{\bf n = 12}[/tex]
