if x+ (1/x) = ϕ ( Golden ratio) ⇒ x^( 2000) + (1/x^( 2000) )=?


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Question Number 190360 by mnjuly1970 last updated on 01/Apr/23

$i f x + \frac{1}{x} = φ (G o l d e n r a t i o)$ $\Rightarrow x^{2000} + \frac{1}{x^{2000}} = ?$
	Answered by Frix last updated on 01/Apr/23

	$x = e^{\pm i \frac{π}{5}}$ $x^{2000} = e^{\pm 400 π i} = 1$ $1 + 1 = 2$
	Commented by mnjuly1970 last updated on 02/Apr/23

	$p l e a s e s o l u t i o n$
	Answered by BaliramKumar last updated on 02/Apr/23

	$x + \frac{1}{x} = \frac{\sqrt{5} + 1}{2} . . . . . . (i)$ $x^{2} + \frac{1}{x^{2}} = \frac{\sqrt{5} - 1}{2} . . . . . . . . (i i)$ $x^{3} + \frac{1}{x^{3}} = \frac{1 - \sqrt{5}}{2} . . . . . . . . . . (i i i)$ $(i i) \times (i i i)$ $(x^{2} + \frac{1}{x^{2}}) (x^{3} + \frac{1}{x^{3}}) = - {(\frac{\sqrt{5} - 1}{2})}^{2}$ $x^{5} + \frac{1}{x^{5}} + x + \frac{1}{x} = \frac{\sqrt{5} - 3}{2}$ $x^{5} + \frac{1}{x^{5}} = \frac{\sqrt{5} - 3}{2} - (x + \frac{1}{x})$ $x^{5} + \frac{1}{x^{5}} = \frac{\sqrt{5} - 3}{2} - (\frac{\sqrt{5} + 1}{2}) = - 2$ $∴ x^{5} = - 1$ $x^{2000} + \frac{1}{x^{2000}} = {(x^{5})}^{400} + \frac{1}{{(x^{5})}^{400}}$ $x^{2000} + \frac{1}{x^{2000}} = {(- 1)}^{400} + \frac{1}{{(- 1)}^{400}} = 2$
	Commented by mnjuly1970 last updated on 02/Apr/23

	$t h x s i r$
	Answered by BaliramKumar last updated on 02/Apr/23

	$x + \frac{1}{x} = φ$ $x + \frac{1}{x} = \frac{\sqrt{5} + 1}{2} [φ = \frac{\sqrt{5} + 1}{2}]$ $x + \frac{1}{x} = 2 c o s (\frac{π}{5}) [c o s (\frac{π}{5}) = \frac{\sqrt{5} + 1}{4}]$ $s q u a r e b o t h s i d e$ $x^{2} + \frac{1}{x^{2}} + 2 = 4 {c o s}^{2} (\frac{π}{5})$ $x^{2} + \frac{1}{x^{2}} = 4 {c o s}^{2} (\frac{π}{5}) - 2 = 2 [2 {c o s}^{2} (\frac{π}{5}) - 1]$ $x^{2} + \frac{1}{x^{2}} = 2 c o s 2 (\frac{π}{5})$ $x^{n} + \frac{1}{x^{n}} = 2 c o s (\frac{n π}{5})$ $x^{2000} + \frac{1}{x^{2000}} = 2 c o s (\frac{2000 π}{5}) = 2 c o s (400 π)$ $x^{2000} + \frac{1}{x^{2000}} = 2 \times 1 = 2 [c o s 2 n π = 1]$ $x^{2023} + \frac{1}{x^{2023}} = 2 c o s (\frac{2023 π}{5})$ $\Rightarrow 2 c o s (404 π + \frac{3 π}{5}) \Rightarrow 2 c o s (\frac{3 π}{5})$ $\Rightarrow 2 c o s (π - \frac{2 π}{5}) \Rightarrow - 2 c o s (\frac{2 π}{5})$ $\Rightarrow - 2 c o s 72 ° = - 2 s i n 18 ° = - 2 (\frac{\sqrt{5} - 1}{4})$ $\Rightarrow - (\frac{\sqrt{5} - 1}{2})$
	Commented by mnjuly1970 last updated on 02/Apr/23

	$t h x v e r y n i c e$
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