For each $c \in [0;2]$ I have to examine monotonicity and limit of the sequence $a_1 = c$, $a_{n+1} = 1 + \frac{(a_n - 1)^2}{17}$. I only solved equation $g = 1 + \frac{(g-1)^2}{17}$ and $g = 1$ or $g = 18$. Can you help me with the rest?
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 $c\in [0;2]$?
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