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Question Number 214186 by hardmath last updated on 30/Nov/24

$$\mathrm{If}\mathrm{x}+\frac{1}{\mathrm{x}}=1\mathrm{find}{\mathrm{x}}^{61}+\frac{1}{{\mathrm{x}}^{61}}+4=?$$

Answered by Rasheed.Sindhi last updated on 30/Nov/24

$$x+\frac{1}{x}=1;x^{61}+\frac{1}{x^{61}}+4=?$$$$x+\frac{1}{x}=1\Rightarrow x^2=x-1$$$$x^3=x^2-x=(x-1)-x=-1$$$${\left(x^3\right)}^{20}={(-1)}^{20}=1\Rightarrow x^{60}=1$$$$$$$$x^{61}+\frac{1}{x^{61}}+4=x^{60}.x+\frac{1}{x^{60}.x}+4$$$$=x+\frac{1}{x}+4=1+4=5$$

Answered by BaliramKumar last updated on 01/Dec/24

$$x+\frac{1}{x}=2cos\left(\frac{\pi }{3}\right)$$$$x^{61}+\frac{1}{x^{61}}=2cos\left(\frac{61\pi }{3}\right)x^n+\frac{1}{x^n}=2cos\left(n\theta \right)$$$$x^{61}+\frac{1}{x^{61}}=2cos\left(\frac{60\pi +\pi }{3}\right)\Rightarrow 2cos(20\pi +\frac{\pi }{3})$$$$x^{61}+\frac{1}{x^{61}}=2cos\left(\frac{\pi }{3}\right)$$$$x^{61}+\frac{1}{x^{61}}=1$$$$x^{61}+\frac{1}{x^{61}}+4=1+4=5$$$$$$

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