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Question Number 152670 by liberty last updated on 31/Aug/21

Answered by Olaf_Thorendsen last updated on 31/Aug/21

$$x_n=x_{n-1}+x_{n-2}$$$$x_n-x_{n-1}-x_{n-2}=0$$$$r^2-r-1=0$$$$r=\frac{1+\sqrt{5}}{2}(=\phi :\mathrm{golden}\mathrm{ratio})$$$$\mathrm{or}r=\frac{1-\sqrt{5}}{2}(={\phi }^{\prime }=-\frac{1}{\phi })$$$$x_n=\lambda {\left(\frac{1+\sqrt{5}}{2}\right)}^n+\mu {\left(\frac{1-\sqrt{5}}{2}\right)}^n$$$$$$$$x_0=\lambda {\left(\frac{1+\sqrt{5}}{2}\right)}^0+\mu {\left(\frac{1-\sqrt{5}}{2}\right)}^0=1$$$$x_0=\lambda +\mu =1\left(1\right)$$$$$$$$x_1=\lambda {\left(\frac{1+\sqrt{5}}{2}\right)}^1+\mu {\left(\frac{1-\sqrt{5}}{2}\right)}^1=1$$$$\lambda \phi -\frac{\mu }{\phi }=1\left(2\right)$$$$$$$$\Rightarrow \lambda =\frac{1}{2}(1+\frac{1}{\sqrt{5}})=\frac{\phi }{\sqrt{5}}$$$$\mathrm{and}\mu =\frac{1}{2}(1-\frac{1}{\sqrt{5}})=-\frac{{\phi }^{\prime }}{\sqrt{5}}$$$$x_n=\frac{\phi }{\sqrt{5}}{\phi }^n+(-\frac{{\phi }^{\prime }}{\sqrt{5}}){\phi }^{\prime n}$$$$x_n=\frac{1}{\sqrt{5}}({\phi }^{n+1}-{\phi }^{\prime n+1})$$

Commented by puissant last updated on 31/Aug/21

$$Binetformula..$$

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