## 2. The Largest of Consecutive Numbers

The sum of 1259 consecutive integers is 1259. What is the largest addend?

Result: $\quad 630$
## First solution.

Let the smallest number be denoted $\mathrm{s} x$.

The next number is $x+1$, then $x+2$, ..., and so on up to the 1259th number which is $x+1258$.

$x+x+1+x+2+\cdots+x+1258=1259 x+1+2+\cdots+1258$

$$
=1259 x+\frac{1258 \cdot 1259}{2}=1259 x+1259 \cdot 629
$$

It holds that:

$1259 x+1259 \cdot 629=1259 /: 1259$

$x+629=1$

$x=-628$

The largest addend is the number $x+1258=-628+1258=630$.