Let and be the roots of 5x^2+6x-2=0.

Question

Let $\alpha$ and $\beta$ be the roots of $5x^2+6x-2=0$. If $S_n=\alpha^n+\beta^n$, $n=1,2,3,\ldots$, then:

Accepted Answer

Since $5\alpha^2+6\alpha-2=0$, multiplying by $\alpha^4$ and adding the analogous $\beta$ relation:

$$5S_6+6S_5=2(\alpha^4+\beta^4)=2S_4.$$

Solutions · 1