Question 22
Let $(a_{n})_{n \geq 0}$ be a sequence of real numbers such that $a_{0} \in \mathbb{R}$ and $a_{n+1}=a_{n}^{2}-8 a_{n}+18, \forall n \geq 0$.
a) Show that if the sequence $(a_{n})_{n \geq 0}$ is convergent, then $\exists p \in \mathbb{N}$ such that $a_{n+1}=a_{n}, \forall n \geq p$.
b) Determine the set of natural values of $a_{0}$ for which the sequence $a_{n}$ is convergent.
a) Show that if the sequence $(a_{n})_{n \geq 0}$ is convergent, then $\exists p \in \mathbb{N}$ such that $a_{n+1}=a_{n}, \forall n \geq p$.
b) Determine the set of natural values of $a_{0}$ for which the sequence $a_{n}$ is convergent.