Question

If \(\omega\) is an imaginary cube root of unity, then the value of \((2-\omega)\left(2-\omega^{2}\right)+2(3-\omega)\left(3-\omega^{2}\right)+\ldots . .+(n-1)(n-\omega)\left(n-\omega^{2}\right)\) is
(A) \(\frac{n^{2}}{4}(n+1)^{2}-n\)
(B) \(\frac{n^{2}}{4}(n+1)^{2}+n\)
(C) \(\frac{n^{2}}{4}(n+1)^{2}\)
(D) \(\frac{n^{2}}{4}(n+1)-n\)

Answer 1

Ans: (A)
\(\sum_{r=2}^{n}(r-1)(r-\omega)\left(r-\omega^{2}\right)=\sum_{r=2}^{n}\left(r^{3}-1\right)=\left[\frac{n^{2}(n+1)^{2}}{4}-1\right]-(n-1)=\frac{n^{2}(n+1)^{2}}{4}-n\)

Answer 2

generic cialis online <a href="https://ordergnonline.com/">cialis 40mg</a> best over the counter ed pills

Answer 3

tadalafil 10mg sale <a href="https://ordergnonline.com/">tadalafil 40mg oral</a> red ed pill