x + (1/x) = 1 x^(100) + (1/x^(100) ) = ? x^(2020) + (1/x^(2020) ) = ?


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Question Number 77591 by naka3546 last updated on 08/Jan/20

$x + \frac{1}{x} = 1$ $x^{100} + \frac{1}{x^{100}} = ?$ $x^{2020} + \frac{1}{x^{2020}} = ?$
Commented by jagoll last updated on 08/Jan/20

$\frac{1}{x^{2}} + x^{2} = - 1$ $\frac{1}{x^{4}} + x^{4} = - 1$ $\frac{1}{x^{8}} + x^{8} = - 1$ $\frac{1}{x^{2 n}} + x^{2 n} = - 1 s o \frac{1}{x^{100}} + x^{100} = - 1$
Commented by abdomathmax last updated on 08/Jan/20

$x + \frac{1}{x} = 1 \Rightarrow x^{2} + 1 = x \Rightarrow x^{2} - x + 1 = 0$ $Δ = 1 - 4 = - 3 \Rightarrow x_{1} = \frac{1 + i \sqrt{3}}{2} = e^{\frac{i π}{3}}$ $x_{2} = \frac{1 - i \sqrt{3}}{2} = e^{- \frac{i π}{3}}$ $x = x_{1} \Rightarrow x^{100} + x^{- 100} = e^{\frac{i 100 π}{3}} + e^{- \frac{i 100 π}{3}}$ $= 2 c o s (\frac{100 π}{3}) = 2 c o s (\frac{99 π + π}{3}) = 2 c o s (33 π + \frac{π}{3})$ $= 2 c o s (π + \frac{π}{3}) = - 2 \times \frac{1}{2} = - 1$ $x = x_{2} \Rightarrow x^{100} + x^{- 100} = e^{- \frac{i 100 π}{3}} + e^{\frac{i 100 π}{3}} = - 1$ $f o r x^{2020} + \frac{1}{x^{2020}}$ $x = x_{1} \Rightarrow x^{2020} + x^{- 2020} = e^{i \frac{2020 π}{3}} + e^{- \frac{i 2020 π}{3}}$ $= 2 c o s (\frac{2020 π}{3}) = 2 c o s (\frac{2019 π + π}{3})$ $= 2 c o s (\frac{2019 π}{3} + \frac{π}{3}) = . . .$
Commented by naka3546 last updated on 08/Jan/20

$n = 6 \Rightarrow t h e a n s w e r i s n o t - 1$
Commented by jagoll last updated on 08/Jan/20

$h o w y o u g e t 6 s i r ?$
Commented by naka3546 last updated on 08/Jan/20

$p r o v e i t i f t h e f o r m u l a i s r i g h t f o r a l l n .$
	Answered by MJS last updated on 08/Jan/20

	$x = \frac{1}{2} \pm \frac{\sqrt{3}}{2} i \Rightarrow \frac{1}{x} = \frac{1}{2} \mp \frac{\sqrt{3}}{2} i$ $z = x^{m} + \frac{1}{x^{m}}$ $x^{6 n} = 1 \Rightarrow z = 2$ $x^{6 n + 1} = x \Rightarrow z = 1$ $x^{6 n + 2} = - \frac{1}{x} = - \bar{x} \Rightarrow z = - 1$ $x^{6 n + 3} = - 1 \Rightarrow z = - 2$ $x^{6 n + 4} = - x \Rightarrow z = - 1$ $x^{6 n + 5} = \frac{1}{x} = \bar{x} \Rightarrow z = 1$ $100 = 16 \times 6 + 4 \Rightarrow z = - 1$ $2020 = 336 \times 6 + 4 \Rightarrow z = - 1$
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