Determine if the following sequence is monotonic Consider the recursion $$ X_1 = 10 \quad \&\quad  X_{n+1} = \sqrt {3+2X_n}$$
Is it monotonic or not?
problem
What I have done so far.
$X_1 = 10$
$X_2 = \sqrt {23}$
$X_3 = \sqrt {2 \sqrt{23} +3}$ 
We can clearly see that the first term is bigger than the second and so on, so the sequence is actually decreasing. A great proof would be to prove that $(X_1-X_2)>0$ But you can't just do that by plugging in numbers.
I tried doing this so far. 
I should get $(X_1-X_2)>0$ but I wind up getting $<0$. What's my mistake? work

HINT: Note that $$x_{n+2}^2-x_{n+1}^2=2(x_{n+1}-x_n).$$ Now factor the LHS and since the sequence is clearly positive both $x_{n+2}-x_{n+1}$ and $x_{n+1}-x_n$ has the same parity for all $n\in \mathbb{N}.$ Then inductively $x_2-x_1$ must have the same parity, check it. For that comparing $x_2^2$ and $x_1^2$ is enough. (Why?)
