Question

Let $z$ be a complex number satisfying $z^2+z+1=0.$ If $n$ is not a multiple of $3,$ then the value of $z^n+z^{2n}$ is equal to

• A. $2$
• B. $-2$
• C. $0$
• D. $-1$

Answer 1

Answer: (D)

$-1$

We have, $z^2+z+1=0$ ...(1)

$\Rightarrow (z-1)(z^2+z+1)=0,$

$\therefore z^3=1.$

If $n$ is not a multiple of $3,$ then we can write $n=3m+r,$ where $m\in I$ and $r=1$ or $2,$

then $2n=6m+2r$

If $r=1,$ then $2r=2$

$\therefore z^n+z^{2n}={\left(z^3\right)}^m.z^r+{\left(z^3\right)}^{2n}.z^{2r}=z^r+z^{2r}$

$=z+z^2=-1$ [Using (1)]

If $r=2,$ then $2r=4$

$\therefore 2n=3(m+1)+1$

$\therefore z^n+z^{2n}={\left(z^3\right)}^m.z^r+{\left(z^3\right)}^{m+1}.z^1=z^2+z=-1$

Hence, $z^n+z^{2n}=-1$