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.
Hint: show by induction that the sequence is nonincreasing and positive.
Hint: show by induction that the sequence is nonincreasing and positive.