For every $c\in [0;2]$ determine whether the sequence $\{a_n\}_{n\geq 1}$ which is defined as follows:
$a_1=c$, $a_{n+1}=1+\frac{(a_n-1)^2}{17}$ for $n\geq 1$
is monotonic for sufficiently large $n$, and determine whether its limit exits and if it exists, give its value.
I have no idea what to do with this problem. I was able to see that $a_{n+1}-a_n$ is a quadratic function and I also found ot that limit, if exists, is equal to either $1$ or $18$ (that's beacause if $a_n$ id convergent to $g$ then every subsequence is also convergent to $g$). So how to determine the limit for every $c\in [0;2]$?
Answer
$$a_{n+1}-a_n=\frac{17+(a_n-1)^2-17a_n}{17}=\frac{(a_n-1)(a_n-18)}{17}$$
This number is $\leq 0$ for $a_n\in[1,2]$ and $>0$ for $a_n\in[0,1)$. But observe that even if $a_0<1$ then $a_2>1$. So, the sequence eventually decreases while it is bounded by $0$.
You already got the possible limits and you can discard $18$ because the sequence is always smaller than say $2$.
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