试证
$$
D_n=\left|\begin{array}{cccccc}
\alpha+\beta & \alpha \beta & 0 & \cdots & 0 & 0 \\
1 & \alpha+\beta & \alpha \beta & \cdots & 0 & 0 \\
0 & 1 & \alpha+\beta & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & \alpha+\beta & \alpha \beta \\
0 & 0 & 0 & \cdots & 1 & \alpha+\beta
\end{array}\right|=\frac{\alpha^{n+1}-\beta^{n+1}}{\alpha-\beta}(\alpha \neq \beta) .
$$