Mathematics

The key is recognizing that $\alpha^r + \beta^r$ follows a recurrence relation you can build from Vieta's formulas. Since $\alpha$ and $\beta$ satisfy the original equation, they obey a linear recurrence—use this to express each power sum in terms of previous ones. Then sum the infinite series by treating it as a geometric-like pattern that telescopes or converges to a closed form.

The key is recognizing that $\alpha^r + \beta^r$ follows a recurrence relation you can build from Vieta's formulas. Since $\alpha$ and $\beta$ satisfy the original equation, they obey a linear recurrence—use this to express each power sum in terms of previous ones. Then sum the infinite series by treating it as a geometric-like pattern that telescopes or converges to a closed form.