Question 17
For each positive integer $n \ge 3$, define $A_n$ and $B_n$ as
$$
A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \dots + \sqrt{n^2 + 2n - 1},
$$
$$
B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \dots + \sqrt{n^2 + 2n}.
$$
Determine all positive integers $n \ge 3$ for which $\lfloor A_n \rfloor = \lfloor B_n \rfloor$.
*Note*. For any real number $x$, $\lfloor x \rfloor$ denotes the largest integer $N$ such that $N \le x$.
$$
A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \dots + \sqrt{n^2 + 2n - 1},
$$
$$
B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \dots + \sqrt{n^2 + 2n}.
$$
Determine all positive integers $n \ge 3$ for which $\lfloor A_n \rfloor = \lfloor B_n \rfloor$.
*Note*. For any real number $x$, $\lfloor x \rfloor$ denotes the largest integer $N$ such that $N \le x$.