Finding the formula for the fibunacci sequence

[Susanne217]

1 If I am given the function \frac{1}{-x^2-x+1} = \sum_{j=0}^{\infty} F_{j} x^{j}

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 F_j = F_{j-1}+F_{j-2}

Homework Equations

The Attempt at a Solution

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


With the inital conditions F_0 = F_1 = 1

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


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

so this gives me F_0 = \alpha_1 + \alpha_2 = 1

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

\alpha_1, \alpha_2 = \pm \frac{1}{\sqrt{5}}

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

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

How is that HallsoftIvy?

Susanne

 

[HallsofIvy]

1 If I am given the function \frac{1}{-x^2-x+1} = \sum_{j=0}^{\infty} F_{j} x^{j}

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 F_j = F_{n-1}+F_{n-2}

You mean F_n= F_{n-1}+ F_{n-2}

Homework Equations

The Attempt at a Solution

I know that the recusive relation can be written as F_n = \alpha_1(r_1)^n + \alpha_2 (r_2)^n

Okay, so its just a matter of finding r_1 and r_2. If F_n= r^n the equation F_n= F_{n-1}+ F_{n-2} becomes r^n= r^{n-1}+ r^{n-2}. Dividing that equation by r^{n-1} gives r^2= r+ 1. What are the roots of that equation?

No fair! You edited and added your solution while I was responding!

 

[Susanne217]

You mean F_n= F_{n-1}+ F_{n-2}


Okay, so its just a matter of finding r_1 and r_2. If F_n= r^n the equation F_n= F_{n-1}+ F_{n-2} becomes r^n= r^{n-1}+ r^{n-2}. Dividing that equation by r^{n-1} gives r^2= r+ 1. What are the roots of that equation?

No fair! You edited and added your solution while I was responding!

Sorry HallsoftIvy :)