(1)
\[
\begin{vmatrix}
1 & 2 & 3 & \cdots & n-1 & n\\
2 & 3 & 4 & \cdots & n & n+1\\
3 & 4 & 5 & \cdots & n+1 & n+2\\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots\\
n & n+1 & n+2 & \cdots & 2n-2 & 2n-1\\
\end{vmatrix}
\]
1
\[
D_n:=
\begin{vmatrix}
1 & 2 & 3 & \cdots & n-1 & n\\
2 & 3 & 4 & \cdots & n & n+1\\
3 & 4 & 5 & \cdots & n+1 & n+2\\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots\\
n & n+1 & n+2 & \cdots & 2n-2 & 2n-1\\
\end{vmatrix}
\]
\[
\xrightarrow[k=n,n-1,n-2,\ldots,2]{r_k-r_{k-1}}
\begin{vmatrix}
1 & 2 & 3 & \cdots & n-1 & n\\
1 & 1 & 1 & \cdots & 1 & 1\\
1 & 1 & 1 & \cdots & 1 & 1\\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots\\
1 & 1 & 1 & \cdots & 1 & 1\\
1 & 1 & 1 & \cdots & 1 & 1\\
1 & 1 & 1 & \cdots & 1 & 1\\
\end{vmatrix}=0.
\]
则, $D_1=1$, $D_2=-1$, $D_n=0$, $\forall\ n=3,4,5,\ldots$.