# UKMT Special (Problem $7$)

A sequence

$b_1, b_2, b_3,...$

of non-zero real numbers have the property that

$b_{n + 2} = \frac{b_{n + 1}^2 - 1}{b_n}$

for all positive integers $n$.

Suppose that $b_1 = 1$ and $b_2 = k$, where $1 < k < 2$. Show that there is some constant $B$, depending on $k$, such that

$- B \leq b_n \leq B$

for all $n$.

Also show that, for some $1 < k < 2$, there is a value of $n$ such that

$b_n > 2020$

[UKMT BMO $2019$ Round $2$, Q$4$]

Note by Yajat Shamji
8 months, 2 weeks ago

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