<s>[INST] Relation between $S_{n-1}$ , $S_n$ and $S_{n+1}$ We know $ \alpha $ and $ \beta $ are roots of $ax^2+bx+c = 0 $.
also $S_{n-1} = \alpha^{n-1} + \beta^{n-1}$ , $S_{n} = \alpha^{n} + \beta^{n}$ and $S_{n+1} = \alpha^{n+1} + \beta^{n+1}$.
How we can find relation between $S_{n-1}$ , $S_n$ and $S_{n+1}$ ?
Note : We know that relation has $a$ , $b$ and $c$.  
 [/INST] $\alpha$ and $\beta$ satisfy the equation:
$ax^2+bx+c=0$
Thus they satisfy the following equation by multiplying with $x^{n-1}$:
$ax^{n+1}+bx^n+cx^{n-1}=0$
Thus:
$a\alpha^{n+1}+b\alpha^n+c\alpha^{n-1}=0$
$a\beta^{n+1}+b\beta^n+c\beta^{n-1}=0$
And by addition:
$aS_{n+1}+bS_n+cS_{n-1}=0$
It generalizes trivially to any polynomial of degree $\geq 2$
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