## Task 2 - 040932

The terms of the following sum are formed according to a certain rule. Find this rule, and calculate $x$ as simply as possible!

$$
x=\frac{6}{5 \cdot 7}+\frac{6}{7 \cdot 9}+\frac{6}{9 \cdot 11}+\frac{6}{11 \cdot 13}+\ldots+\frac{6}{31 \cdot 33}
$$
First, we factor out 3. Each of the summands of the form $\frac{2}{a(a+2)}$ can be written as the difference of two fractions:

$$
\frac{2}{a(a+2)}=\frac{1}{a}-\frac{1}{a+2}
$$

Therefore, the sum to be calculated is

$$
x=3\left(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9} \pm \ldots-\frac{1}{31}+\frac{1}{31}-\frac{1}{33}\right)=3 \cdot \frac{28}{165}=\frac{28}{55}
$$