Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

$r^{th}termofthegivenseries=r\left[\right(r+1)-\omega ]\left[\right(r+1)-{\omega }^2]=r\left[\right(r+1{)}^2-(\omega +{\omega }^2)(r+1)+{\omega }^3]=r[(r+1{)}^2-(-1\left)\right(r+1)+1]=r\left[\right(r^2+3r+3\left)\right]=r^3+3r^2+3rThussumofthegivenseries={\sum }_{r_1}^{(n-1)}(r^3+3r^2+3r)=\frac{1}{4}(n-4{)}^2{n}^2+3.\frac{1}{6}(n-1\left)\right(n\left)\right(2n-1)+3.\frac{1}{2}(n-1)n=\frac{1}{4}(n-1\left)\right(n^2+3n+4)$