<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] Note that $$\alpha^{n} + \beta^{n}=(\alpha + \beta)(\alpha^{n-1} + \beta^{n-1})-\alpha\beta\cdot(\alpha^{n-2} + \beta^{n-2})$$
Now using the relation between the coefficients of a quadratic equation and its roots, we get that
$$S_n=-\frac{b}{a}S_{n-1}-\frac{c}{a}S_{n-2}$$
Or, in other words, we can write that $$S_{n+1}=-\frac{b}{a}S_{n}-\frac{c}{a}S_{n-1}$$
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