Solving simple problems 4

A sequence $u_n$ is determined by the relations

$u_1 = \alpha + \beta$

$u_n = \alpha + \beta - \frac{\alpha \beta}{u_{n - 1}}$


Show that

$u_n = \frac{\alpha^{n+ 1} - \beta^{n + 1}}{\alpha^n - \beta^n}$

$ = \alpha^n + \beta^n $


solution

u_n = u_{n - 1} + 

$u_1 = \alpha + \beta$

$\alpha + \beta + \left( \alpha + \beta - \frac{\alpha \beta}{\alpha + \beta} \right) + \ ...$

$\alpha + \beta + ( \frac{    (\alpha + \beta)^2 - \alpha \beta    }{\alpha + \beta + \ ...}$