x-1-x-1-x-1907-1-x-1907-

Question Number 102771 by bramlex last updated on 11/Jul/20

x + \frac{1}{x} = - 1 \Rightarrow x^{1907} + \frac{1}{x^{1907}} ?

Answered by ajfour last updated on 11/Jul/20

x^{2} + x + 1 = 0

\Rightarrow x = ω, ω^{2}

ω^{1907} + \frac{1}{ω^{1907}} = \frac{1}{ω} + ω = ω^{2} + ω = - 1 .

Answered by floor(10²Eta[1]) last updated on 11/Jul/20

x^{2} = - x - 1

x^{4} = x^{2} + 2 x + 1 = (- x - 1) + 2 x + 1 = x

\Rightarrow x^{4} = x \Rightarrow x^{16} = x^{4} = x

x^{160} = x^{10} = x^{4} . x^{4} . x^{2} = x . x . x^{2} = x^{4} = x

\Rightarrow x^{160} = x \Rightarrow x^{1600} = x^{10} = x

★ x^{307} = {(x^{10})}^{30} . x^{7} = x^{30} . x^{4} . x^{3} = {(x^{10})}^{3} . x . x^{3}

= x^{3} . x^{4} = x^{3} . x = x

\Rightarrow x^{1907} = x^{1600} . x^{307} = x . x = x^{2}

x^{1907} + \frac{1}{x^{1907}} = x^{2} + \frac{1}{x^{2}} = \frac{x^{4} + 1}{x^{2}} = \frac{x + 1}{- x - 1} = - 1

Commented by Rasheed.Sindhi last updated on 11/Jul/20

C o o l!

Commented by bramlex last updated on 11/Jul/20

jooss \dots

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