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Pe: 2015-09-04T13:17:55+03:00 In: In:

Numere complexe

Buna ziua ! va rog sa ma ajutati cu cateva indicii la doua exercitii

1. Calculati (1/2 + (i rad 3)/2)^2014 + (1/2 – (i rad 3)/2)^2014
2. Daca x este numar real nenul , si x+ 1/x numar intreg , demonstrati ca x^n + 1/x^n este numar intreg , n numar natural nenul.

va multumesc mult ! 😳

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2 raspunsuri

  1. A_Cristian
    expert (VI)

    1. Foloseste forma trigonometrica a unui numar complex.
    2. Intuitia ar trebui sa-ti spuna ca se rezolva prin inductie.

    De exemplu:
    Fie\;k=x+\frac{1}{x} \\ k^2=(x+\frac{1}{x})^2=x^2+\frac{1}{x^2}+2 \Rightarrow x^2+\frac{1}{x^2} = k^2-2

  2. gunty
    maestru (V)

    1. Utilizam formula Moivre…

        \[ \begin{array}{l} \left( {\frac{1}{2} + i \cdot \frac{{\sqrt 3 }}{2}} \right)^{2014} + \left( {\frac{1}{2} - i \cdot \frac{{\sqrt 3 }}{2}} \right)^{2014} = \left( {\cos \frac{\pi }{3} + i \cdot \sin \frac{\pi }{3}} \right)^{2014} + \left( {\cos \frac{\pi }{3} - i \cdot \sin \frac{\pi }{3}} \right)^{2014} = \cos \frac{{2014\pi }}{3} + i \cdot \sin \frac{{2014\pi }}{3} + \cos \frac{{2014\pi }}{3} - i \cdot \sin \frac{{2014\pi }}{3} = \\ = 2\cos \left( {670\pi + \frac{{4\pi }}{3}} \right) = 2\cos \frac{{4\pi }}{3} = 2\cos \left( {\pi + \frac{\pi }{3}} \right) = - 2\cos \frac{\pi }{3} = - 1. \\ \end{array} \]

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