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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 30, 2019
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Here is my understanding of this:
In order to get the 2nd term, or 2000, we have the following:
2000 = x * y - 1. If we set y = 3000, then: x =2001/3000 =0.667
So, the sequence would look like this:
0.667, 2000, 3,000, 1.5005, 0.0008335, 0.667, 2000, 3000.....etc. Then 3000 repeats every 5th term. 
Any other value for x will not have 3000 in the sequence as far as I understand it.
Note: Somebody should check this. Thanks.

 Oct 1, 2019

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