Question

If $z+z^{-1}=1$, then $z^{100}+z^{-100}$ is equal to

• A. I
• B. -I
• C. 1
• D. -1

Answer 1

The correct option is D -1

$z+\frac{1}{2}=1$ $z^2-z+1=0$
$z^{100}+\frac{1}{z^{100}}$ $\therefore z=\frac{1\pm \sqrt{1-4}}{2}=\frac{1\pm i\sqrt{3}}{2}$
$=(-w{)}^{100}+\frac{1}{(-w{)}^{100}}$ $z=-\left(\frac{-1+i\sqrt{3}}{2}\right),-\left(\frac{-1-i\sqrt{3}}{2}\right)$
$=w^{100}+\frac{1}{w^{100}}$ $z=-w,-w^2$
$=w+w^2$
$=-1$