5
$\begingroup$

Consider the sequence $\{a_n\}_{n\in\Bbb N}$ where $a_n = \lfloor n \sqrt2 \rfloor + \lfloor n \sqrt3 \rfloor $. Show that there exists infinitely many odd and even integers in this sequence.

  • 0
    Hint: The parity of $a_n$ is the same as $b_n=\lfloor n \sqrt3 \rfloor - \lfloor n \sqrt2 \rfloor$. Now think about the possible values of $b_{n+1}-b_n$.2016-04-13

2 Answers 2