Question

Ex ăsta vă rog….. vreau rezolvare completă

Answer 1

Răspuns:

A

Explicație pas cu pas:

$\boxed{ \frac{1}{(3n - 2)(3n + 1)} = \frac{1}{3} \cdot \Big(\frac{1}{3n - 2} - \frac{1}{3n + 1}\Big)}$

$S_{n} = \frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 11} + ... + \frac{1}{(3n - 2)(3n + 1)} =$

$= \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{4}\Big) + \frac{1}{3} \cdot \Big(\frac{1}{4} - \frac{1}{7}\Big) + \frac{1}{3} \cdot \Big(\frac{1}{7} - \frac{1}{11}\Big) + ... + \frac{1}{3} \cdot \Big(\frac{1}{3n - 2} - \frac{1}{3n + 1}\Big)$

$= \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{4} + \frac{1}{4} - \frac{1}{7} + \frac{1}{7} - \frac{1}{11} + ... + \frac{1}{3n - 2} - \frac{1}{3n + 1}\Big)$

$= \frac{1}{3} \cdot \Big(\frac{1}{1} - \frac{1}{3n + 1}\Big) = \frac{1}{3} \cdot \frac{3n + 1 - 1}{3n + 1}$

$= \frac{1}{3} \cdot \frac{3n}{3n + 1}</p><p>= \red {\bf \frac{n}{3n + 1}}$