Calculating the determinant of this matrix Given this (very) tricky determinant, how can we calculate it easily?
$$\begin{pmatrix} \alpha + \beta & \alpha \beta & 0 & ... & ... & 0 \\ 1 & \alpha + \beta &  \alpha \beta & 0 & ... & 0 \\ 0 & 1 & \alpha + \beta & \alpha \beta & ... & ... \\ ... & ... & ... & ... & ... & 0 \\ ... & ... & .... & ... & ... & \alpha \beta \\ 0 & 0 & 0 & ... & 1 & \alpha + \beta   \\ \end{pmatrix} \in M_{n\times n}$$
EDIT:

I have to prove it is equal to $\frac{{\alpha}^{n+1} - {\beta}^{n+1}}{\alpha - \beta}$

Any help is appreciated, I just could not find a trick to ease it up!

Let $D_n$ represent the determinant of this $n\times n$ matrix.  Expanding on the first column, we see that $D_n=(\alpha+\beta)D_{n-1}-\alpha\beta D_{n-2}$, where the second is found by expanding on the first row of the resulting minor.  The recurrence begins with $D_1=\alpha+\beta, D_2=(\alpha+\beta)^2-\alpha\beta=\alpha^2+\alpha\beta+\beta^2$.
To prove that this recurrence equals $\frac{\alpha^{n+1}-\beta^{n+1}}{\alpha-\beta}$ one can use induction.
