Question 13
Let $a_1, a_2, a_3, \dots$ be a sequence of positive integers such that $a_1 = 2021$ and
$$
\sqrt{a_{n+1} - a_n} = \lfloor \sqrt{a_n} \rfloor.
$$
Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence.
$$
\sqrt{a_{n+1} - a_n} = \lfloor \sqrt{a_n} \rfloor.
$$
Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence.