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Question Number 128498 by mathmax by abdo last updated on 07/Jan/21

$$\mathrm{simplify}{\mathrm{A}}_{\mathrm{n}}=\frac{1}{{(1+\sqrt{3}\mathrm{i})}^{\mathrm{n}}}+\frac{1}{{(1-\sqrt{3}\mathrm{i})}^{\mathrm{n}}}$$

Answered by TheSupreme last updated on 08/Jan/21

$$1+i\sqrt{3}=2e^{i\frac{\pi }{3}}$$$$A_n=\frac{1}{2^n}{e}^{-i\frac{n\pi }{3}}+\frac{1}{2^n}{e}^{i\frac{n\pi }{3}}=$$$$=\frac{1}{2^n}\left(2cos\left(\frac{\pi }{3}\right)\right)=\frac{1}{2^n}$$

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