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Consider sequences of positive real numbers of the form x, 2000, y ..., in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x does the term 3000 appear somewhere in the sequence?

 Sep 22, 2019
 #1
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If I understand your question correctly, the "generating function" is as follows:

 

a(n) =[a(n - 1) * a(n - 2)] - 1. So, in order to get 3,000 for the 3rd term, you have the following:

x * 2,000 - 1 = 3,000

x =3,001 / 2,000

x =1.5005 - This is the only positive real number that will give 3,000 as the 3rd term. After that the sequence "blows up"!, in the sense that the 4th term will be : 2,000 x 3,000 - 1 =5,999,999......and so on.

 Sep 22, 2019
 #2
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Wait that's not right the generating function is \(a(n)=[a(n-1)*a(n+1)]-1\)

For example in the context of the problem, 2000=xy-1. That's what the problem means when it says the two neighboring numbers

Moondust  Sep 30, 2019
edited by Guest  Sep 30, 2019

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