First slide

Algebra of complex numbers

Question

$The value of the expression$ $\begin{array}{r} (1 + \frac{1}{ω}) (1 + \frac{1}{ω^{2}}) + (2 + \frac{1}{ω}) (2 + \frac{1}{ω^{2}}) \end{array}$ $+ (3 + \frac{1}{ω}) (3 + \frac{1}{ω^{2}}) + \dots . + (n + \frac{1}{ω}) (n + \frac{1}{ω^{2}})$ $where ω is an imaginary cube root of unity, is$

Moderate

A

$\frac{n (n^{2} - 2)}{3}$
B

$\frac{n (n^{2} + 2)}{3}$
C

$\frac{n (n^{2} - 1)}{3}$
D

None of these

Solution

$W e h a v e \sum_{k = 1}^{n} (k + \frac{1}{ω}) (k + \frac{1}{ω^{2}}) = \sum_{k = 1}^{n} (k + ω^{2}) (k + ω) (∵ ω^{3} = 1) = \sum_{k = 1}^{n} (k^{2} + k (ω + ω^{2}) + ω^{3}) = \sum_{k = 1}^{n} (k^{2} - k + 1) (∵ 1 + ω + ω^{2} = 0) = \frac{n (n + 1) (2 n + 1)}{6} - \frac{n (n + 1)}{2} + n = \frac{n (n^{2} + 2)}{3}$

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