6.342 Let $S_{n}=\alpha^{n}+\beta^{n}$, where $\alpha$ and $\beta$ are the roots of the equation $a x^{2}+b x+c=0$. Find the relationship between $S_{n}, S_{n+1}, S_{n+2}$.
## Solution.

$$
\begin{gathered}
S_{n+1} \cdot S_{1}=\left(\alpha^{n+1}+\beta^{n+1}\right)(\alpha+\beta)=\left(\alpha^{n+2}+\beta^{n+2}\right)+\alpha \beta\left(\alpha^{n}+\beta^{n}\right)=S_{n+2}+\alpha \beta \cdot S_{n} \\
\text { Since } S_{1}=-\frac{b}{a}, \alpha \beta=\frac{c}{a} \Rightarrow S_{n+1} \cdot\left(-\frac{b}{a}\right)=S_{n+2}+\frac{c}{a} \cdot S_{n}
\end{gathered}
$$

Answer: $\quad S_{n+2}=\frac{-\left(b \cdot S_{n+1}+c \cdot S_{n}\right)}{a}$.