About the sequence satisfying $a_n=a_{n-1}a_{n+1}-1$ "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 2001 appear somewhere in the sequence?
(A) 1 (B) 2 (C) 3 (D) 4 (E) More than 4"
Can anyone suggest a systematic way to solve this problem? Thanks!

We are given that $xy=2001$.  Clearly if one of them is $1$ and the other is $2001$, then $2001$ appears.  Try the other factorizations to see if $2001$ appears.
