Question

The value of the expression $1\cdot (2-\omega )(2-{\omega }^2)+2\cdot (3-\omega )(3-{\omega }^2)+.....+(n-1)(n-\omega )(n-{\omega }^2)$, 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

Answer: (B)

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

$rth$ term of the given series can be written as:

$T_r=(r-1)(r-\omega )(r-{\omega }^2)$

$\therefore$expression= $\sum _{r=2}^n(r-1)(r-\omega )(r-{\omega }^2)$

=$\sum _{r=2}^n(r-1)(r+r^2+1)$

=$\sum _{r=2}^n{r}^3-1$

=$(2^3+3^3+\cdots +n^3)-(n-1)$

=${\left(\frac{n(n+1)}{2}\right)}^2-n$