Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
be a positive integer and let
![\{a_n\}](/media/m/2/5/2/252d82e82000c8ff418959c98eeed9e9.png)
be defined by
![a_0 = 0](/media/m/0/4/7/0471eca23d871c11edd98d44cf6ecffb.png)
and
![a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).](/media/m/f/0/7/f07261d0848f8765dd71aef130ad797f.png)
Show that for each positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
,
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
is a positive integer.
%V0
Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and
$$a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).$$
Show that for each positive integer $n$, $a_n$ is a positive integer.