How can one show that $(U_n)$ is arithmetic progression

Question

Given that $(U_n)$ a numerical sequence such that :

$U_0$=3

$U_{n+1}=\sqrt{(U_n)^2+8n+16}$

Show that $(U_n)$ is a arithmetic progression.

So I have to show that $U_{n+1}-U_n=r$ where $r$ is a constant.

I started with this :

$(U_{n+1})^2-U_n^2=8n+16$

So $U_{n+1}-U_n=\frac{8(n+2)}{U_{n+1}+U_n}$

Also , for $n=0$ we have

$U_1=\sqrt{9+16}=5$

And $U_0=3$

Answer 1

hint: $U_0 = 3, U_1 = 5, U_2 = 7 \Rightarrow U_n = 2n+3\Rightarrow U_{n+1} = ? \Rightarrow U_{n+1}-U_n = ?$

Answer 2

By definition, we have $U_{n + 1} = |U_n + 4| = U_n + 4$, since $U_0 = 3 > 0$. We clearly have $U_n = 4n + 3$. This is an arithmetic progression, with $d = U_{n + 1} - U_n = 4$.