First slide

Algebra of complex numbers

Question

If $x^{2} + x + 1 = 0$ , then the value of ${(x + \frac{1}{x})}^{2} + {(x^{2} + \frac{1}{x^{2}})}^{2} + {(x^{3} + \frac{1}{x^{3}})}^{2} + \dots + {(x^{27} + \frac{1}{x^{27}})}^{2}$ is

Moderate

A

27
B

72
C

45
D

54

Solution

$\begin{array}{l} x^{2} + x + 1 = 0 \\ \Rightarrow x = ω or ω^{2} \end{array}$
Let x = $ω$ . Then,
$\begin{matrix} x + \frac{1}{x} = ω + \frac{1}{ω} = ω + ω^{2} = - 1 \\ x^{2} + \frac{1}{x^{2}} = ω^{2} + \frac{1}{ω^{2}} = ω^{2} + ω = - 1 \\ x^{3} + \frac{1}{x^{3}} = ω^{3} + \frac{1}{ω^{3}} = 2 \\ x^{4} + \frac{1}{x^{4}} = ω^{4} + \frac{1}{ω^{4}} = ω + ω^{2} = - 1, etc. \end{matrix}$
$∴ {(x + \frac{1}{x})}^{2} + {(x^{2} + \frac{1}{x^{2}})}^{2} + {(x^{3} + \frac{1}{x^{3}})}^{2} + \dots + {(x^{27} + \frac{1}{x^{27}})}^{2}$
$= [{(x + \frac{1}{x})}^{2} + {(x^{2} + \frac{1}{x^{2}})}^{2} + {(x^{4} + \frac{1}{x^{4}})}^{2} + \dots + {(x^{26} + \frac{1}{x^{26}})}^{2}] + [{(x^{3} + \frac{1}{x^{3}})}^{2} + {(x^{6} + \frac{1}{x^{6}})}^{2} + {(x^{9} + \frac{1}{x^{9}})}^{2} + \dots + {(x^{27} + \frac{1}{x^{27}})}^{2}] = 18 + 9 (2^{2}) = 54$

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