First slide

Algebra of complex numbers

Question

Let $ω \neq 1$ , be a cube root of unity, and $f : I \to C$ be defined by $f (n) = 1 + ω^{n} + ω^{2 n}$ then range of $f$ is

Moderate

A

$\{0\}$
B

$\{0, 3\}$
C

$\{0, 1, 3\}$
D

$\{0, 1\}$

Solution

If $n = 3 k, k \in I$ then $f (n) = 1 + 1 + 1 = 3$
If $n = 3 k + 1, k \in I$ then $f (n) = 1 + ω + ω^{2} = 0$
If $n = 3 k + 2 k \in I,$ then $f (n) = 1 + w^{2} + w = 0$
Thus, range of f is $\{0, 3\} .$

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