2002 AMC 12B Problems/Problem 13

Problem

The sum of $18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is

Let $a, a+1, \ldots, a + 17$ be the consecutive positive integers. Their sum, $18a + \frac{17(18)}{2} = 9(2a+17)$ , is a perfect square. Since $9$ is a perfect square, it follows that $2a + 17$ is a perfect square. The smallest possible such perfect square is $25$ when $a = 4$ , and the sum is $225 \Rightarrow \mathrm{(B)}$ .