x-1-x-1-x-100-1-x-100-x-2020-1-x-2020-

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}) = \dots

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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