Arithmetic sequence of natural numbers

Question

Consider an arithmetic progression of natural numbers with a non-zero common difference. For each of the members of the progression its square root is taken, and if the square root is not an integer, it is rounded (this is the function of rounding - (x) ) to the nearest integer(and (2.5)=3, for example ). Could it be that all the non-integer square roots of this progression are rounded to the same side(either larger or smaller)? How do I solve this one? I'm really clueless. I got that the progression is rising. Please help.:)

Answer 1

Hint: Suppose that the common difference is $d$. Consider some large square $N^2$, where $N$ is not divisible by $d$, and the two elements of the sequence closest to it.