Question

If $n$ is not a multiple of 3 then the value of ${\omega }^n+{\omega }^{2n}=$

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

The correct option is D $-1$

If $n$ is not a multiple of 3, the $n$ can be written as
$n=\{3k+1,3k+2\};$ k is any integer
Case (1)
$n=3k+1$
${\omega }^n+{\omega }^{2n}={\omega }^{3k+1}+{\omega }^{6k+2}$
$={\omega }^{3k}.\omega +{\omega }^{6k}.{\omega }^2$
$=\omega +{\omega }^2$
$=-1$
Case (2)
$n=3k+2$
${\omega }^n+{\omega }^{2n}={\omega }^{3k+2}+{\omega }^{6k+4}$
$={\omega }^{3k}.{\omega }^2+{\omega }^{6k}.{\omega }^4$
$={\omega }^2+{\omega }^4$
$={\omega }^2+\omega$
$=-1$
In both cases,
${\omega }^n+{\omega }^{2n}=-1;n\notin$ multiple of 3