## 5. Integers

Let $x_{1}, x_{2}, x_{3}, x_{4}, \ldots, x_{1013}$ be consecutive integers, in increasing order. If

$$
-x_{1}+x_{2}-x_{3}+x_{4}-\cdots-x_{1011}+x_{1012}-x_{1013}=1013
$$

determine the absolute value of the number $x_{1013}$.

Result: $\quad 507$
## Solution.

From $x_{1}$ to $x_{1012}$, we have 1012 numbers, or 506 pairs, so the equality from the problem can be written as

$$
\left(-x_{1}+x_{2}\right)+\left(-x_{3}+x_{4}\right)+\cdots+\left(-x_{1011}+x_{1012}\right)-x_{1013}=1013
$$

Consecutive integers differ by 1, so the value of each parenthesis is 1.

There are 506 parentheses, so we have $506 - x_{1013} = 1013$.

Therefore, $x_{1013} = -507$, or $\left|x_{1013}\right| = 507$.