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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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You could solve the constant-coefficient linear difference equation explicitly. This is second-order, so your characteristic polynomial is quadratic... –  J. M. Oct 13 '11 at 12:35
    
@J.M.: Thanks. I am being silly. –  olga Oct 13 '11 at 12:45
    
See this answer for a closed formula to solve such equations. –  Pierre-Yves Gaillard Oct 13 '11 at 13:14

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