# Let $a_0=a_1=1$. Define $a_{n+1}=\frac{1+a_n+a_{n-1}^2}{4}$, does it converge?

Let $$a_0=a_1=1$$. Define $$a_{n+1}=\frac{1+a_n+a_{n-1}^2}{4}$$. Does the sequence {$$a_n$$} converge to a finite limit as $$n \to \infty$$? Find this limit if it does?

Putting in successive value of n shows the value decreases as n increases, so it must converge. And then we can solve for $$l=\frac{1+l+l^2}{4}$$, to get the limit. Now I don't know how to formally prove that it converges.

• Since induction shows all terms are $\le1$, if the limit exists it's $L:=(3-\sqrt{5})/2$. Study the behaviour of $\epsilon_n:=a_n-L$.
– J.G.
Oct 9 '20 at 11:58