Let and be the roots of x^2-6x-2=0.

Question

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$. If $a_n=\alpha^n-\beta^n$ for $n\ge1$, then the value of $\dfrac{a_{10}-2a_8}{2a_9}$ is equal to:

Accepted Answer

As in Q162's method, $a_8(\alpha^2-2)-\ldots$ using $\alpha^2-2=6\alpha,\ \beta^2-2=6\beta$:

$$\frac{a_{10}-2a_8}{2a_9}=\frac{6(a_9)}{2a_9}=3.$$

Solutions · 1