Răspuns:
A
Explicație pas cu pas:
[tex]\boxed{ \frac{1}{(3n - 2)(3n + 1)} = \frac{1}{3} \cdot \Big(\frac{1}{3n - 2} - \frac{1}{3n + 1}\Big)} \\ [/tex]
[tex]S_{n} = \frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 11} + ... + \frac{1}{(3n - 2)(3n + 1)} = \\ [/tex]
[tex]= \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{4}\Big) + \frac{1}{3} \cdot \Big(\frac{1}{4} - \frac{1}{7}\Big) + \frac{1}{3} \cdot \Big(\frac{1}{7} - \frac{1}{11}\Big) + ... + \frac{1}{3} \cdot \Big(\frac{1}{3n - 2} - \frac{1}{3n + 1}\Big) \\ [/tex]
[tex]= \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{4} + \frac{1}{4} - \frac{1}{7} + \frac{1}{7} - \frac{1}{11} + ... + \frac{1}{3n - 2} - \frac{1}{3n + 1}\Big) \\[/tex]
[tex]= \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{3n + 1}\Big) = \frac{1}{3} \cdot \frac{3n + 1 - 1}{3n + 1} \\ [/tex]
[tex]= \frac{1}{3} \cdot \frac{3n}{3n + 1}
= \red {\bf \frac{n}{3n + 1}} \\ [/tex]
