Question

The value of the expression $1\cdot (2-\omega )(2-{\omega }^2)+2\cdot (3-\omega )(3-{\omega }^2)+....+(n-1)\cdot (n-\omega )(n-{\omega }^2)$, where $\omega$ is an imaginary cube root of unity is

• A. ${\left\{\frac{n(n+1)}{2}\right\}}^2$
• 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 ${\left\{\frac{n(n+1)}{2}\right\}}^2-n$

Let $S_n=1\cdot (2-\omega )(2-{\omega }^2)+2\cdot (3-\omega )(3-{\omega }^2)+....+(n-1)\cdot (n-\omega )(n-{\omega }^2)$
$T_n=(n-1)(n-\omega )(n-{\omega }^2)$
$=n^3-n^2({\omega }^2+\omega +1)+n({\omega }^3+\omega +\omega )-{\omega }^3$
As we know, $({\omega }^2+\omega +1)=0,{\omega }^3=1$
$T_n=n^3-1$
$S_n=\sum _{n=2}^n(n^3-1)$
$=\sum _{n=1}^n{n}^3-\sum _{n=1}^{n}1-(1^3-1)$
$={\left\{\frac{n(n+1)}{2}\right\}}^2-n$