How Can I Solve a Determinant Problem Using Induction?

[kidsmoker]

Homework Statement

Hi,
i'm trying to solve this problem:

http://img4.imageshack.us/img4/3876/53065718.jpg .[/URL]

The Attempt at a Solution

I have shown it for n=2 and n=3 then I was going to use induction to prove it for all n, but I can't seem to find a way to do it. Please help!

Thanks.

 

[CompuChip]

If you expand the determinant along the last column, then you will get terms of the form
a_i^n \begin{vmatrix} <br /> 1 &amp; a_2 &amp; a_2^2 &amp; \cdots &amp; a_2^{n-1} \\ <br /> 1 &amp; a_3 &amp; a_3^2 &amp; \cdots &amp; a_3^{n-1} \\ <br /> \vdots &amp; \cdots &amp; \cdots &amp; \cdots &amp; \cdots \\<br /> 1 &amp; a_{n} &amp; a_n^2 &amp; \cdots &amp; a_n^{n-1} \\ <br /> \end{vmatrix}