Solve $n$th order determinant Solve $n$th order determinant:
$$
    \begin{pmatrix}
    1 & 2 & 3 & \cdots & n-2 & n-1 &n\\
    2 & 3 & 4 & \cdots & n-1& n& 1\\
    3 & 4 & 5 & \cdots & n &1 &2\\
    \vdots& \vdots& \vdots& \ddots& \vdots& \vdots&\vdots\\
    n-2 &n-1&n &\cdots &n-5 &n-4 &n-3\\
    n-1 &n&1 &\cdots &n-4 &n-3 &n-2\\
    n &1&2 &\cdots &n-3 &n-2 &n-1\\
    \end{pmatrix}
$$
I tried adding $2$nd to $n$th row to the $1$st row, then the first row is:
$$
   \begin{pmatrix}
   \frac{n(n+1)}{2} &\frac{n(n+1)}{2} \cdots \frac{n(n+1)}{2}\\
   \end{pmatrix}
$$
then I took out $\frac{n(n+1)}{2}$ from the first row to get 
$$
   \begin{pmatrix}
   1 & 1 & \cdots& 1\\
   \end{pmatrix}
$$
and then tried to work with that but I don't think this is the correct approach because anything I tried doesn't look nice. 

With the help of WA, we get this sequence
$$
1, -3, -18, 160, 1875, -27216, -470596, \dots
$$
oeis/A052182 tells us that the determinant is $(-1)^{n-1}n^{n-2}\dfrac{(n^2 + n)}{2}$, except for the sign.
