Is there an effective way of finding a particular $x_n$, say $x_5$, of a system of difference equations $x_{n}=ax_{n+1}+bx_{n-1}$ where $a, b$ are constants and the $n$'s say are $\leq k$ (apart from actually substituting each equation into the next)? Thanks.
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2You could solve the constant-coefficient linear difference equation explicitly. This is second-order, so your characteristic polynomial is quadratic... – J. M. ain't a mathematician Oct 13 '11 at 12:35
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@J.M.: Thanks. I am being silly. – olga Oct 13 '11 at 12:45
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See this answer for a closed formula to solve such equations. – Pierre-Yves Gaillard Oct 13 '11 at 13:14
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