Complex Numbers 5 Question 14
15. The value of the expression
$1(2-\omega)\left(2-\omega^{2}\right)+2(3-\omega)\left(3-\omega^{2}\right)+\ldots$
$+(n-1) \cdot(n-\omega)\left(n-\omega^{2}\right)$,
where, $\omega$ is an imaginary cube root of unity, is….
(1996, 2M)
True/False
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Answer:
Correct Answer: 15. $\frac{n(n+1)}{2}{ }^{2}-n$
Solution:
- Here, $\left.T _r=(r-1)(r-\omega)(r-\omega)^{2}\right]=\left(r^{3}-1\right)$
$$ \therefore \quad S _n=\sum _{r=1}^{n}\left(r^{3}-1\right)=\frac{n(n+1)}{2}^{2}-n $$
