证明: $D_n=\left|\begin{array}{ccccc}a+b & a b & \cdots & 0 & 0 \\ 1 & a+b & \cdots & 0 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & a+b & a b \\ 0 & 0 & \cdots & 1 & a+b\end{array}\right|=\left\{\begin{array}{ll}\frac{a^{n+1}-b^{n+1}}{a-b}, & a \neq b \\ (n+1) a^n, & a=b\end{array}\right.$.