Question

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

• A. $\frac{n(n+1)}{2}$
• B. ${\left\{\frac{n(n+1)}{2}\right\}}^2-n$
• C. ${\frac{n(n+1)}{4}}^2$
• D. $\frac{n(n+1)}{2n1}$

Answer 1

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

$(2-w)(2-w^2)=(4+1-2w-2w^2)=(7-2(1+w+w^2\left)\right)=7=2^3-1$
$2(3-w)(3-w^2)=2(9+1-3w-3w^2)=2(13-3(1+w+w^2\left)\right)=26=3^3-1$
Similarly we get,
$(n-1)(n-w)(n-w^2)=(n-1)(n^2+1-nw-n{w}^2)=n^3-1$
By adding them we get,
$1^3-1+2^3-1+3^3-1+......+n^3-1$=$\frac{n^2{(n+1)}^2}{4}-n$