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Question Number 136537 by Jamshidbek last updated on 23/Mar/21

Answered by Olaf last updated on 23/Mar/21

$$a+\frac{1}{a}=-1$$$$a^2+a+1=0$$$$a=\frac{-1\pm i\sqrt{3}}{2}=e^{i\frac{2\pi }{3}}\mathrm{or}{e}^{i\frac{4\pi }{3}}$$$$1+2+3+4+5+6+7+8+9+1+0+1+1$$$$=\frac{9\times 10}{2}+3=48=3n(n=16)$$$$\Rightarrow 1234567891011=3p,p\in \mathbb{N}$$$$\mathrm{and}1110987654321=3q,q\in \mathbb{N}$$$$\mathrm{If}a=e^{i\frac{2\pi }{3}}:$$$$a^{1234567891011}+\frac{1}{a^{1110987654321}}$$$$=a^{i\frac{2\pi }{3}\times 3p}+\frac{1}{e^{i\frac{2\pi }{3}\times 3q}}=e^{2p\pi i}+e^{-2q\pi i}=2$$$$\mathrm{If}a=e^{i\frac{4\pi }{3}}:$$$$=a^{i\frac{4\pi }{3}\times 3p}+\frac{1}{e^{i\frac{4\pi }{3}\times 3q}}=e^{4p\pi i}+e^{-4q\pi i}=2$$$$\mathrm{answer}\mathrm{is}:2$$

Commented by Jamshidbek last updated on 23/Mar/21

$$thankyousir$$

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