计算
$$
\begin{aligned}
&D_n=\left|\begin{array}{ccccc}
\alpha+\beta & \alpha \beta & 0 & \cdots & 0 \\
1 & \alpha+\beta & \alpha \beta & \cdots & 0 \\
0 & 1 & \alpha+\beta & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
0 & 0 & 0 & \cdots & \alpha+\beta
\end{array}\right|_{(n)}
\end{aligned}
$$