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Question Number 19435 by Tinkutara last updated on 11/Aug/17

$$\mathrm{If}\alpha =\cos \frac{2\pi }{5}+i\sin \frac{2\pi }{5},\mathrm{then}\mathrm{find}\mathrm{the}$$$$\mathrm{value}\mathrm{of}\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4.$$

Answered by ajfour last updated on 11/Aug/17

$$\underset{\mathrm{r}=1}{\sum ^4}{\alpha }^{\mathrm{r}}=\frac{\alpha (1-{\alpha }^4)}{1-\alpha }=\frac{{\mathrm{e}}^{\mathrm{i}2\pi /5}-{\mathrm{e}}^{\mathrm{i}2\pi }}{1-{\mathrm{e}}^{\mathrm{i}2\pi /5}}=-1.$$$$$$

Commented by Tinkutara last updated on 11/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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