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?
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.