Question

If $\alpha, \beta$ are the roots of the quadratic equation $x^2 - 2x + 4 = 0$, then the value of $\alpha^n + \beta^n$ is:

• A. $2^{n+1} \cos\left(\frac{n\pi}{3}\right)$
• B. $2^{n+1} \sin\left(\frac{n\pi}{3}\right)$
• C. $2^n \cos\left(\frac{n\pi}{3}\right)$
• D. $2^n \sin\left(\frac{n\pi}{3}\right)$

Hint:

Test for $n = 1$: $\alpha^1 + \beta^1$ is the sum of the roots, which is $2$. Plugging $n = 1$ into option (A) gives $2^{1+1} \cos(\pi/3) = 4 \times 0.5 = 2$, validating the answer instantly.

Answer 1

The Correct Option is A