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**valerian** $\displaystyle x_0 > x_1$ and r and s are positive numbers with p + q = 1. $\displaystyle x_n = px_{n-1} + qx_{n-2}$ for all $\displaystyle n \geq 2$. Prove that $\displaystyle (x_n)$ is convergent and find limits in terms of $\displaystyle x_0, x_1$, p and q.

Okay I began by letting p = 1 - q so I could eliminate that variable. Then I began working it out up to $\displaystyle x_7$ but I couldn't find any pattern in terms of coefficients of the polynomial. It's a polynomial with respect to p. I am trying to get this to work out to something that I can put into summation notation, prove it is monotone, and then find an upper and/or lower bound but I am unsure how to go about putting it in summation notation - it's entirely possible this is not the way to do it but it seems most logical to me. The first term (in order of descending exponent on p) of the expansion is always $\displaystyle p^{n-1}*a_1$. The last term is $\displaystyle a_0$ when n is even and $\displaystyle a_1$ when n is odd. These are the only real patterns I've been able to discern though. Any insight is much appreciated. Thanks.