Question

$S=\{x:x=\frac{1}{\sqrt{5}}({\left(\frac{1+\sqrt{5}}{2}\right)}^n-{\left(\frac{1-\sqrt{5}}{2}\right)}^n)\}$ if $n=3$ then find the solution.

Answer 1

$\begin{array}{c}S=\left\{x=x=\frac{1}{\sqrt{5}}\left({\left(\frac{1+\sqrt{5}}{2}\right)}^n-{\left(\frac{1-\sqrt{5}}{2}\right)}^n\right)\right\}\\ n=3\\ \Rightarrow S=\frac{1}{\sqrt{5}}\left({\left(\frac{1+\sqrt{5}}{2}\right)}^3-{\left(\frac{1-\sqrt{5}}{2}\right)}^3\right)\\ \Rightarrow S=\frac{1}{\sqrt{5}}\left\{\left(\frac{2\sqrt{5}}{2}\right)\times \left(\frac{1+2\sqrt{5}+5}{4}+\frac{1-5}{4}+\frac{1-2\sqrt{5}+5}{4}\right)\right\}\\ \Rightarrow S=\frac{1}{\sqrt{5}}\left\{\sqrt{5}\left(\frac{6+2\sqrt{5}}{4}-1+\frac{6-2\sqrt{5}}{4}\right)\right\}\\ S=\frac{\sqrt{5}}{\sqrt{5}}\left\{\left[3-1\right]\right\}=2\end{array}$