计算行列式 $D_{n+1}=\left|\begin{array}{ccccc}a_1^n & a_1^{n-1} b_1 & \cdots & a_1 b_1^{n-1} & b_1^n \\ a_2^n & a_2^{n-1} b_2 & \cdots & a_2 b_2^{n-1} & b_2^n \\ \vdots & \vdots & & \vdots & \vdots \\ a_{n+1}^n & a_{n+1}^{n-1} b_{n+1} & \cdots & a_{n+1} b_{n+1}^{n-1} & b_{n+1}^n\end{array}\right|$ ，其中 $a_1 a_2 \cdots a_{n+1} \neq 0$ ．