I am not sure which method to use here. Should I do it for $n=2$ and $n=3$ and then use induction on $n$?
Let $\alpha_1,\alpha_2,\ldots,\alpha_n \in \mathbb{R}$, where $n \geq 2$. Show that $$\left|\begin{array}{ccccc} 1 & \alpha_1 & \alpha_1^2 & \cdots & \alpha_1^{n-1}\\ 1 & \alpha_2 & \alpha_2^2 & \cdots & \alpha_2^{n-1}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & \alpha_n & \alpha_n^2 & \cdots & \alpha_n^{n-1} \end{array} \right| = \prod_{1 \leq i < j \leq n}(\alpha_j-\alpha_i).$$
Nice to see that Warwick are still using this question (link is broken now). In my year most people heard through others that this is called the Vandermonde matrix, and a bunch of proofs for it are found online. I realise that the deadline is today, but do attempt to understand the proofs.
I found the first proof the easiest to understand at the time, but perhaps you'd like to use the second proof to help with your induction.
Good luck!