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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.

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    $\begingroup$ Since induction shows all terms are $\le1$, if the limit exists it's $L:=(3-\sqrt{5})/2$. Study the behaviour of $\epsilon_n:=a_n-L$. $\endgroup$
    – J.G.
    Oct 9 '20 at 11:58
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Hint: show by induction that the sequence is nonincreasing and positive.

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  • $\begingroup$ Could the downvoter please explain his or her reasons ? $\endgroup$ Oct 10 '20 at 8:48

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