Question

If $n$ is a positive integer, then$(1+i\sqrt{3}{)}^n+(1-i\sqrt{3}{)}^n$ is equal to

• A. $2^{n-1}cos\frac{n\mathrm{\pi }}{3}$
• B. $2^{n}cos\frac{n\mathrm{\pi }}{3}$
• C. $2^{n+1}cos\frac{n\mathrm{\pi }}{3}$
• D. None of these

Answer 1

The correct option is C

$2^{n+1}cos\frac{n\mathrm{\pi }}{3}$

Explanation for correct option

We know that, As per below formula

$(1+i\sqrt{3})=2(\cos \frac{\mathrm{\pi }}{3}+i\sin \frac{\mathrm{\pi }}{3})$,

$(1-\sqrt{3}i)=2(\cos \frac{\mathrm{\pi }}{3}-i\sin \frac{\mathrm{\pi }}{3})$

From Euler's theorem we know that ${\left(\cos \left(\theta \right)+i\sin \left(\theta \right)\right)}^n=\cos \left(n\theta \right)+i\sin \left(n\theta \right)$

${\therefore (1+i\sqrt{3})}^n+{(1–i\sqrt{3})}^n=2^n(\cos \frac{n\mathrm{\pi }}{3}+i\sin \frac{n\mathrm{\pi }}{3})+2^n(\cos \frac{n\mathrm{\pi }}{3}-i\sin \frac{n\mathrm{\pi }}{3})$

$=2^{n+1}\cos \frac{n\mathrm{\pi }}{3}$

Hence option C is the correct answer.