Question

If $n$ is an integer which leaves remainder one when divided by three, then ${\left(-1+\sqrt{3}i\right)}^n+{\left(-1-\sqrt{3}i\right)}^n$

• A. $-2^{n+1}$
• B. $2^{n+1}$
• C. $-{\left(-2\right)}^n$
• D. $-2^n$

Answer 1

Answer: (D)

$-2^n$

Multiply and divide the given equation by $2^n$:

$z=2^n[\frac{{\left(-1+\sqrt{3}i\right)}^n}{2^n}+\frac{{(-1-\sqrt{3}i)}^n}{2^n}]=2^n[w^n+w^{2n}]$

$=2^n[-1]=-2^n$........[$1+w^n+w^{2n}=0$ for $n$ is not a multiple of $3$]