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 $4$ ]

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Comments

You have until $1^{st}$ December, $3:00$ pm!

Yajat Shamji - 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

UKMT Special (Problem 777)

Comments

UKMT Special (Problem $7$ )