(adsbygoogle = window.adsbygoogle || []).push({}); 1 If I am given the function [tex]\frac{1}{-x^2-x+1} = \sum_{j=0}^{\infty} F_{j} x^{j}[/tex]

Which discribes a sequence of fibunacci numbers 1,1,2,3,5,8,13,21......

Find the formula for the fibunacci sequence for n>=2 and where [tex]F_j = F_{j-1}+F_{j-2}[/tex]

2. Relevant equations

3. The attempt at a solution

I know that the recusive relation can be written as [tex]F_j = \alpha_1(r_1)^j + \alpha_2 (r_2)^j[/tex]

With the inital conditions [tex]F_0 = F_1 = 1[/tex]

Since the poles of the function are [tex]r_1 = \frac{\sqrt{5}-1}{2}[/tex] and [tex]r_2 = \frac{-(\sqrt{5}+1)}{2}[/tex]

which gives me the expression [tex]F_j = \alpha_1(\frac{\sqrt{5}-1}{2})^n + \alpha_2 (\frac{-(\sqrt{5}+1)}{2}})^n[/tex]

so this gives me [tex]F_0 = \alpha_1 + \alpha_2 = 1[/tex]

and [tex]F_1 = \alpha_1(\frac{\sqrt{5}-1}{2}) + \alpha_2 (\frac{-(\sqrt{5}+1)}{2}}) = 1[/tex] and I end up with

[tex]\alpha_1, \alpha_2 = \pm \frac{1}{\sqrt{5}}[/tex]

and this j >= 2 then the formula for the jth fibunacci number must

[tex]F_j = \frac{1}{\sqrt{5}}(\frac{\sqrt{5}+1}{2})^{j+1} -\frac{1}{\sqrt{5}} (\frac{-(\sqrt{5}+1)}{2}})^{j+1}[/tex]

How is that HallsoftIvy?

Susanne

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# Homework Help: Finding the formula for the fibunacci sequence

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