The following sequence oscillates
$$a_1 = 1$$ $$a_{n+1} = -(a_n)^2+4a_n-1$$
If its possible to find the limit of a function that oscillates between 2 and 3 (after n = 2), and if so how do we find it?
2026-03-29 23:39:57.1774827597
Recursive sequence problem (while it oscillates)
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Well of you prove that the limit exists and its value is $L$, then by continuity you have $$L=-L^2+4L-1$$ I.e. $L$ is one of the two solutions of the equation, namely $$\frac{3\pm\sqrt{5}}{2}.$$ Again, you can do this after showing that the limit exists.
However in your case (i.e. with your initial value 1) the sequence is $1,2,3,2,3,2,3,...$ therefore it does not converge. To prove this, you have $$\lim\sup a_n=3\mbox{ and }\lim\inf a_n=2$$ And because there exist two subsequences (odd and even) converging to different values, the limit does not exist.