Question

The value of the expression $2(1+\omega )(1+{\omega }^2)+3(2\omega +1)(2{\omega }^2+1)+4(3\omega +1)(3{\omega }^2+1)+...$

$+(n+1)(n\omega +1)(n{\omega }^2+1)$ is
($\omega$ is the cube root of unity)

• A. $\frac{n^2{(n+1)}^2}{4}$
• B. ${\left(\frac{n(n+1)}{2}\right)}^2+n$
• C. ${\left(\frac{n(n+1)}{2}\right)}^2-n$
• D. None of the above

Answer 1

The correct option is B ${\left(\frac{n(n+1)}{2}\right)}^2+n$

The $r^{th}$ term in the expression $(r+1)(1+r\omega )(1+r{\omega }^2)$
$=(1+r)(1+r^2+r\omega +r{\omega }^2)$
$=(1+r)(1+r^2-r)$
$=(r^3+1)$
Hence, the sum of the series $=\sum _{r=1}^n{r}^3+1$
Sum $=\frac{\left(n{)}^2\right(n+1{)}^2}{4}+n$
Hence, option B is correct.