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# algebra

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If $$x + \frac{1}{x} = -1,$$ find $$x^{99} + \frac{1}{x^{99}}$$

Jul 8, 2020

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Lemma: If $$x + \dfrac1x = 2\cos \alpha$$, then $$x^n + \dfrac1{x^n} = 2\cos n\alpha$$

If you want to see the proof of this, go here. I have done the proof in a previous answer before.

Now, if we consider $$\alpha = \dfrac{2\pi}3$$ and $$n = 99$$, using the formula directly gives

$$x^{99} + \dfrac1{x^{99}} = 2\cos\left(\dfrac{2\pi}3\cdot 99\right) = \boxed{2}$$

Jul 8, 2020