UKMT Special (Problem 77)

A sequence

b1,b2,b3,...b_1, b_2, b_3,...

of non-zero real numbers have the property that

bn+2=bn+121bnb_{n + 2} = \frac{b_{n + 1}^2 - 1}{b_n}

for all positive integers nn.

Suppose that b1=1b_1 = 1 and b2=kb_2 = k, where 1<k<21 < k < 2. Show that there is some constant BB, depending on kk, such that

BbnB- B \leq b_n \leq B

for all nn.

Also show that, for some 1<k<21 < k < 2, there is a value of nn such that

bn>2020b_n > 2020

[UKMT BMO 20192019 Round 22, Q44]

Note by Yajat Shamji
8 months, 2 weeks ago

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You have until 1st1^{st} December, 3:003:00pm!

Yajat Shamji - 8 months, 2 weeks ago

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