이차방정식 근의 거듭제곱 합

이차방정식 $x ^{2} - x - 1 = 0$의 근과 계수의 관계에서 $\alpha + \beta = 1$, $\alpha \beta = - 1$ $\alpha ^{3} + \beta ^{3}$$= \left( \alpha + \beta \right) ^{3} - 3 \alpha \beta \left( \alpha + \beta \right)$ $= 1 ^{3} - 3 \times ( - 1 ) \times 1 = 4$ $\alpha ^{6} + \beta ^{6}$$= \left( \alpha ^{3} + \beta ^{3} \right) ^{2} - 2 \alpha ^{2} \beta ^{2}$ $= 4 ^{2} - 2 \times ( - 1 ) ^{3} = 18$ (ⅰ) $k = 1$일 때 $\displaystyle \alpha _{3} = \frac{1}{2} \left( \alpha ^{3} + \beta ^{3} \right)$$\displaystyle = \frac{1}{2} \times 4 = 2$ (ⅱ) $k = 2$일 때 $\displaystyle \alpha _{6} = \frac{1}{2} \left( \alpha ^{6} + \beta ^{6} \right)$$\displaystyle = \frac{1}{2} \times 18 = 9$ (ⅲ) $k = 3$일 때 $\alpha _{9}$$\displaystyle = \frac{1}{2} \left( \alpha ^{9} + \beta ^{9} \right)$ $\displaystyle = \frac{1}{2} \left\{ \left( \alpha ^{3} + \beta ^{3} \right) \left( \alpha ^{6} + \beta ^{6} \right) - \alpha ^{3} \beta ^{3} \left( \alpha ^{3} + \beta ^{3} \right) \right\}$ $\displaystyle = \frac{1}{2} \left\{ 4 \times 18 - ( - 1 ) \times 4 \right\}$ $= 38$ (ⅰ), (ⅱ), (ⅲ)에서 $\displaystyle \sum\limits _{k = 1} ^{3} a _{3 k} = a _{3} + a _{6} + a _{9} = 2 + 9 + 38 = 49$