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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?

matrex
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1 Answers1

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hint: prove:

$1 < a_{n+1} < 18 $

(it's actually smaller than 18 but it doesn't matter)

then use induction to prove:

$a_{n+1} < a_n$

if a series is monotonic and blocked, then it has a limit. (let's call it $L$):

$L = 1 + \frac{(L - 1)^2}{17}$

$17L = 17 + L^2 -2L + 1$

$(L-1)(L-18) = 0$

finally, $L=1$ (since we proved $L<18$)

alonkol
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