Complex Numbers — a typical IITJEE Main problem

Example 1.

If \omega is the imaginary cube root of unity, then value of the expression

1(2-\omega)(2-\omega^{2}) + 2(3-\omega)(3-\omega^{2})+\ldots+(n-1)(n-\omega)(n-\omega^{2}) is

(a) \frac{1}{4}{n^{2}}(n+1)^{2}-n

(b) \frac{1}{4}{n^{2}}(n-1)^{2}+n

(c) \frac{1}{4}{n^{2}}(n+1)-n

(d) \frac{1}{4}n(n+1)^{2}-n

Answer. (a).


rth term of the given expression is


because x^{3}-1=(x-1)(x-\omega)(x-\omega^{2}).

Thus, the value of the expression is given by


=2^{3}+3^{3}+ \ldots + n^{3}-(n-1)

=1^{3}+2^{3}+\ldots + n^{3}-n


More later,

Nalin Pithwa

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