# Task 6. (14 points)

Determine the sign of the number

$$
A=\frac{1}{1}-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+\ldots+\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}-\frac{1}{2015}+\frac{1}{2016}
$$

The signs are arranged as follows: “+" before the first fraction, then two “-” and two “+" alternately. The last fraction has a "+" before it.
# Solution

Let's divide all numbers into groups of four numbers:

$$
\left(\frac{1}{4 k+1}-\frac{1}{4 k+2}-\frac{1}{4 k+3}+\frac{1}{4 k+4}\right)
$$

The sum of the numbers in each group is positive:

$$
\begin{gathered}
\left(\frac{1}{4 k+1}-\frac{1}{4 k+2}-\frac{1}{4 k+3}+\frac{1}{4 k+4}\right)>0 \Leftrightarrow \\
\Leftrightarrow \frac{1}{4 k+1}-\frac{1}{4 k+2}>\frac{1}{4 k+3}-\frac{1}{4 k+4} \Leftrightarrow \frac{1}{(4 k+1)(4 k+2)}>\frac{1}{(4 k+3)(4 k+4)}
\end{gathered}
$$

Therefore, the number $A$ is also positive.

Answer: $A>0$