Let \(\alpha\) and \(\beta\) be the roots of the equation \(ax^2+bx+c=0\) and \(p_n = \alpha^n+\beta^n\), where \(n>1\). What is \(ap_{n+1}+bp_n+cp_{n-1}\) equal to?
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Since \(\alpha\) satisfies \(a\alpha^2+b\alpha+c=0\), multiplying by \(\alpha^{n-1}\) gives \(a\alpha^{n+1}+b\alpha^n+c\alpha^{n-1}=0\); the same holds for \(\beta\). Adding the two relations gives \(a(\alpha^{n+1}+\beta^{n+1})+b(\alpha^n+\beta^n)+c(\alpha^{n-1}+\beta^{n-1}) = ap_{n+1}+bp_n+cp_{n-1} = 0\).
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