Question

Let $\omega$ is an imaginary cube roots of unity then the value of $2(\omega +1)({\omega }^2+1)+3(2\omega +1)(2{\omega }^2+1)+........+(n+1)(n\omega +1)(n{\omega }^2+1)$is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$2(\omega +1)({\omega }^2+1)+3(2\omega +1)(2{\omega }^2+1)+........+(n+1)(n\omega +1)(n{\omega }^2+1)={\sum }_{r=1}^n(r+1)(r\omega +1)(r{\omega }^2+1)={\sum }_{r=1}^n(r+1)(r^2{\omega }^3+r\omega +r{\omega }^2+1)={\sum }_{r=1}^n(r+1)(r^2-r+1)=={\sum }_{r-1}^n(r^3+r+r^2-r+1)={\sum }_{r=1}^n\left(r^3\right)+{\sum }_{r-1}^n\left(1\right)={\left[\frac{n(n+1)}{2}\right]}^2+n$.