证明：$n$ 阶行列式

$$
D_n=\left|\begin{array}{cccccc}
2 a & 1 & 0 & \cdots & 0 & 0 \\
a^2 & 2 a & 1 & \cdots & 0 & 0 \\
0 & a^2 & 2 a & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 2 a & 1 \\
0 & 0 & 0 & \cdots & a^2 & 2 a
\end{array}\right|=(n+1) a^n
$$