Question

If $\omega$ is a non-real cube root of unity, then the value of $1\cdot (2-\omega )(2-{\omega }^2)+2\cdot (3-\omega )(3-{\omega }^2)+\cdots$ sum upto $19$ terms is

• A. $56680$
• B. $44080$
• C. $84390$
• D. $64580$

Answer 1

The correct option is B $44080$

$r^{\text{th}}$ term of the given series is
$=r\left[\right(r+1)-\omega ]\left[\right(r+1)-{\omega }^2]$
$=r\left[\right(r+1{)}^2-(\omega +{\omega }^2)(r+1)+{\omega }^3]$
$=r\left[\right(r+1{)}^2-(-1)(r+1)+1]$
$=r[r^2+3r+3]=r^3+3r^2+3r$

Thus, sum of the given series
$S=\sum _{r=1}^{\left(n\right)}(r^3+3r^2+3r)$
$=\frac{1}{4}(n+1{)}^2{n}^2+3\cdot \frac{1}{6}(n+1\left)n\right(2n+1)+3\cdot \frac{1}{2}(n+1)n$
$=\frac{1}{4}(n+1)n(n^2+5n+8)$
Given $n=19$
$\therefore S=\frac{1}{4}\cdot 19\cdot 20\cdot 464=44080$