<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] \begin{align*}
  \alpha^n+\beta^n &=
  (\alpha+\beta)(\alpha^{n-1}+\beta^{n-1})-
  \alpha \beta (\alpha^{n-2}+\beta^{n-2}) \\
  &= (\alpha+\beta) S_{n-1}-\alpha \beta S_{n-2} \\
  S_{n} &= -\frac{b}{a}S_{n-1}-\frac{c}{a}S_{n-2}
\end{align*}
</s>