Question

Let $z$ be a complex number satisfying $z+z^{-1}=1$. A possible value of $n$ when $z^n+z^{-n}$ is minimum, is

• A. $12$
• B. $15$
• C. $10$
• D. $11$

Answer 1

The correct option is B $15$

$z+z^{-1}=1\Rightarrow z^2-z+1=0\Rightarrow z=\frac{1\pm \sqrt{3}i}{2}=-\omega ,-{\omega }^2$
where $\omega$ is the cube root of unity.

Now, $z^n+z^{-n}$
$=(-\omega {)}^n+(-\omega {)}^{-n}\text{or}(-{\omega }^2{)}^n+(-{\omega }^2{)}^{-n}=(-1{)}^n\left[\right(\omega {)}^n+{\left(\frac{1}{\omega }\right)}^n]\text{or}(-1{)}^n\left[\right({\omega }^2{)}^n+{\left(\frac{1}{{\omega }^2}\right)}^n]=(-1{)}^n[{\omega }^n+{\omega }^{2n}]\text{or}(-1{)}^n[{\omega }^{2n}+{\omega }^n]=(-1{)}^n[{\omega }^n+{\omega }^{2n}]$

When $n=3m,m\in Z$
$z^n+z^{-n}=(-1{)}^{3m}[1+1]=(-1{)}^m\times 2$

When $n$ is not a multiple of $3$, then
$z^n+z^{-n}=(-1{)}^n[-1]=(-1{)}^{n+1}$
Therefore, the minimum value of $z^n+z^{-n}$ occurs when $m=1,3,5,\cdots$
Hence, a possible value of $n$ is $15$.