If $\alpha, \beta$ are the roots of $x^2-2 x+4=0$, for $n \in \mathbf{N}$, what is the value of $\alpha^n+\beta^n=$

Question

If $\alpha, \beta$ are the roots of $x^2-2 x+4=0$, for $n \in \mathbf{N}$, what is the value of $\alpha^n+\beta^n=$, $2^{n+2} \cos \left(\frac{n \pi}{3}\right)$, $2^{n+1} \cos \left(\frac{n \pi}{3}\right)$, $2^{n+1} \cos \left(\frac{n \pi}{6}\right)$, $2^{n+2} \cos \left(\frac{n \pi}{6}\right)$

MARKS App · Accepted Answer

Given, $\alpha, \beta$ are roots of $x^2-2 x+4=0$ $\begin{aligned} & x=\frac{2 \pm \sqrt{4-4 \cdot 1 \cdot 4}}{2 \cdot 1} \ & x=\frac{2 \pm \sqrt{-12}}{2} \ & x=1 \pm \sqrt{3 i} \ & x=1+\sqrt{3 i}(o r) 1-\sqrt{3 i} \ & x=2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)=2\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) \ & \alpha=2 \cdot e^{\frac{i \pi}{3}} \end{aligned}$ Similarly, $\beta=2 \cdot e^{\frac{-i \pi}{3}}$ $\begin{aligned} \alpha^n+\beta^n & =\left(2 e^{\frac{i \pi}{3}}\right)^n+\left(2 e^{\frac{-i \pi}{3}}\right)^n=2^n\left(e^{\frac{i n \pi}{3}}+e^{\frac{-i n \pi}{3}}\right) \ & =2^n\left(\cos \frac{n \pi}{3}+i \sin \frac{n \pi}{3}+\cos \frac{n \pi}{3}-i \sin \frac{n \pi}{3}\right) \ & =2^n 2 \cos \frac{n \pi}{3}=2^{n+1} \cdot \cos \frac{n \pi}{3} \end{aligned}$ Hence, option (b) is correct.

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