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Question Number 89852 by M±th+et£s last updated on 19/Apr/20

Answered by mr W last updated on 19/Apr/20

$$F_n=\frac{1}{\sqrt{5}}({\phi }^n-{\psi }^n)with$$$$\phi =\frac{1+\sqrt{5}}{2},\psi =\frac{1-\sqrt{5}}{2}=1-\phi =-\frac{1}{\phi }$$$$x^n{F}_n=\frac{1}{\sqrt{5}}[{\left(x\phi \right)}^n-{\left(x\psi \right)}^n]$$$$\underset{n=1}{\sum ^{\mathrm{\infty }}}{x}^n{F}_n=\frac{1}{\sqrt{5}}[\underset{n=1}{\sum ^{\mathrm{\infty }}}{\left(x\phi \right)}^n-\underset{n=1}{\sum ^{\mathrm{\infty }}}{\left(x\psi \right)}^n]$$$$=\frac{1}{\sqrt{5}}[\frac{x\phi }{1-x\phi }-\frac{x\psi }{1-x\psi }]$$$$=\frac{x}{\sqrt{5}}[\frac{\phi }{1-x\phi }+\frac{1}{\phi +x}]=1$$$$\Rightarrow \frac{\phi }{\frac{1}{x}-\phi }+\frac{1}{\frac{\phi }{x}+1}=\sqrt{5}$$$$lett=\frac{1}{x}$$$$\Rightarrow \frac{\phi }{t-\phi }+\frac{1}{\phi t+1}=\sqrt{5}$$$$t^2+\frac{\sqrt{5}(1-{\phi }^2)-1-{\phi }^2}{\sqrt{5}\phi }t-1=0$$$$t^2-\frac{(\sqrt{5}+1)\phi +2}{\sqrt{5}\phi }t-1=0$$$$t^2-2t-1=0$$$$\Rightarrow t=1+\sqrt{2}(x>0)$$$$\Rightarrow x=\frac{1}{t}=\frac{1}{1+\sqrt{2}}=\sqrt{2}-1$$

Commented by M±th+et£s last updated on 19/Apr/20

$$niceworkthankyousir$$

Commented by mr W last updated on 19/Apr/20

$$answercorrect?$$$$ididitinhurry,notverycarefully$$$$andnotchecked.$$

Commented by M±th+et£s last updated on 19/Apr/20

$$yessiritscorrect$$

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