+26393 If
\(x + \dfrac{1}{x} = \dfrac{1 + \sqrt{5}}{2}\), then find \(x^{2000} + \dfrac{1}{x^{2000} }\).
\(\text{Let Golden ratio $\varphi= \dfrac{1 + \sqrt{5}}{2}$ } \\ \text{Let $\varphi^2= 1+\varphi $ } \)
\(\begin{array}{|rcll|} \hline \mathbf{x + \dfrac{1}{x}} &=& \mathbf{ \varphi } \\ \hline x^{2} + \dfrac{1}{x^{2}} &=& \left(x + \dfrac{1}{x} \right)^2 - 2\\ &=& \varphi^2 - 2 \quad &|\quad \varphi^2= 1+\varphi \\ &=& 1+\varphi - 2 \\ \mathbf{x^{2} + \dfrac{1}{x^{2}}} &=& \mathbf{\varphi - 1} \\ \hline x^{3} + \dfrac{1}{x^{3}} &=& \left(x^{2} + \dfrac{1}{x^{2}}\right)\times \left(x + \dfrac{1}{x}\right) - \left(x + \dfrac{1}{x }\right)\\ &=& (\varphi-1 )\times \varphi - \varphi \\ &=& \varphi^2-\varphi - \varphi \quad &|\quad \varphi^2= 1+\varphi \\ &=& 1+\varphi-2\varphi \\ \mathbf{x^{3} + \dfrac{1}{x^{3}}} &=& \mathbf{1-\varphi} \\ \hline x^{5} + \dfrac{1}{x^{5}} &=& \left(x^{2} + \dfrac{1}{x^{2}}\right)\times \left(x^3 + \dfrac{1}{x^3}\right) - \left(x + \dfrac{1}{x }\right)\\ &=& (\varphi-1 )\times (1-\varphi ) - \varphi \\ &=& \varphi^2-\varphi - \varphi \\ &=& \varphi- \varphi^2 -1 + \varphi- \varphi \\ &=& \varphi- \varphi^2 -1 \quad &|\quad \varphi^2= 1+\varphi \\ &=& \varphi- (1+\varphi ) -1 \\ &=& \varphi- 1 -\varphi -1 \\ \mathbf{x^{5} + \dfrac{1}{x^{5}}} &=& \mathbf{-2} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline x^{10} + \dfrac{1}{x^{10}} &=& \left(x^{5} + \dfrac{1}{x^{5}} \right)^2 - 2\\ &=& \left(-2 \right)^2 - 2\\ &=& 4 - 2\\ \mathbf{x^{10} + \dfrac{1}{x^{10}}} &=& \mathbf{2} \\ \hline x^{20} + \dfrac{1}{x^{20}} &=& \left(x^{10} + \dfrac{1}{x^{10}} \right)^2 - 2\\ &=& 2^2 - 2\\ &=& 4 - 2\\ \mathbf{x^{20} + \dfrac{1}{x^{20}}} &=& \mathbf{2} \\ \hline x^{30} + \dfrac{1}{x^{30}} &=& \left(x^{20} + \dfrac{1}{x^{20}}\right)\times \left(x^{10} + \dfrac{1}{x^{10}}\right) - \left(x^{10} + \dfrac{1}{x^{10} }\right)\\ &=& 2\times 2 - 2 \\ &=& 4 - 2\\ \mathbf{x^{30} + \dfrac{1}{x^{30}}} &=& \mathbf{2} \\ \hline x^{40} + \dfrac{1}{x^{40}} &=& \left(x^{30} + \dfrac{1}{x^{30}}\right)\times \left(x^{10} + \dfrac{1}{x^{10}}\right) - \left(x^{20} + \dfrac{1}{x^{20} }\right)\\ &=& 2\times 2 - 2 \\ &=& 4 - 2\\ \mathbf{x^{40} + \dfrac{1}{x^{40}}} &=& \mathbf{2} \\ \hline x^{50} + \dfrac{1}{x^{50}} &=& \left(x^{40} + \dfrac{1}{x^{40}}\right)\times \left(x^{10} + \dfrac{1}{x^{10}}\right) - \left(x^{30} + \dfrac{1}{x^{30} }\right)\\ &=& 2\times 2 - 2 \\ &=& 4 - 2\\ \mathbf{x^{50} + \dfrac{1}{x^{50}}} &=& \mathbf{2} \\ \hline \ldots \\ \hline \mathbf{x^{2000} + \dfrac{1}{x^{2000}}} &=& \mathbf{2} \\ \hline \end{array}\)
