For integers $n \geq 2$ and $0 \leq k \leq n-2$, compute the determinant$$ \left|\begin{array}{ccccc} 1^{k} & 2^{k} & 3^{k} & \cdots & n^{k} \\ 2^{k} & 3^{k} & 4^{k} & \cdots & (n+1)^{k} \\ 3^{k} & 4^{k} & 5^{k} & \cdots & (n+2)^{k} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ n^{k} & (n+1)^{k} & (n+2)^{k} & \cdots & (2 n-1)^{k} \end{array}\right| . $$
The polynomials $P_{j}(x)=(x+j)^{k}, j=0,1, \ldots, n-1$, lie in the $(k+1)$-dimensional real vector space of polynomials of degree at most $k$. Because $k+1<n$, they are linearly dependent. The columns consist of the evaluations of these polynomials at $1,2, \ldots, n$, so the columns are linearly dependent. It follows that the determinant is zero.