OlympiadMediumAlgebra10–12

If $\alpha$ and $\beta$ are the roots of $x^{2} - x - 1 = 0$ , what is $\alpha^{5} + \beta^{5}$ ?

Se $\alpha$ e $\beta$ são as raízes de $x^{2} - x - 1 = 0$ , quanto vale $\alpha^{5} + \beta^{5}$ ?

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Solution

Let $s_{n} = \alpha^{n} + \beta^{n}$ . From the equation, $s_{n} = s_{n-1} + s_{n-2}$ with $s_{0} = 2$ and $s_{1} = 1$ : $s_{2} = 3$ , $s_{3} = 4$ , $s_{4} = 7$ , $s_{5} = 11$ .

If α\alphaα and β\betaβ are the roots of x2−x−1=0x^{2} - x - 1 = 0x2−x−1=0, what is α5+β5\alpha^{5} + \beta^{5}α5+β5?

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If $\alpha$ and $\beta$ are the roots of $x^{2} - x - 1 = 0$ , what is $\alpha^{5} + \beta^{5}$ ?