Fibonacci formula?

1. Aug 31, 2008

brandy

im just curious. is there a formula for the fibonacci formula in terms of..well terms. like the nth term =..?
iv been trying to figure it out for a couple of days now but am not that smart.

2. Aug 31, 2008

Santa1

3. Aug 31, 2008

Defennder

You could derive it, if you know enough elementary linear algebra and in particular diagonalisation of matrices. It's not that difficult. You start off with recursive definition of the n+1 and nth term and n-1 term, put them all into a matrix and show that it is diagonalisable, then write out the matrix equation.

4. Sep 2, 2008

brandy

done the algebra but have only learnt +-x/ matrices.
how does the n+1 thing work
like i said am not that smart.

5. Sep 2, 2008

Defennder

There are probably other ways to derive it, but I'm only familiar with the one with matrices. There's a current thread on this here:
https://www.physicsforums.com/showthread.php?p=1856158

6. Sep 2, 2008

HallsofIvy

Staff Emeritus
The terms in a Fibonacci sequence obey the recursive rule Fn+2= Fn+1+ Fn. One common way of solving such equations is to try a solution of the form Fn= an. Then Fn+1= an+1 and Fn+2= an+2 so the equation becomes an+2= an+1+ an. Dividing by an gives a2= a+ 1 or a2- a- 1= 0. Solving that by the quadratic formula,
$$a= \frac{1\pm\sqrt{5}}{2}$$
In other words,
$$F_n= \left(\frac{1+\sqrt{5}}{2}\right)^n$$
and
$$F_n= \left(\frac{1-\sqrt{5}}{2}\right)^n$$
both satisfy Fn+2= Fn+1+ Fn.

Since that is a linear equation, any solution of that equation can be written
$$A\left(\frac{1+\sqrt{5}}{2}\right)^n+ B\left(\frac{1-\sqrt{5}}{2}\right)^n$$

Now, looking at the first two terms of the Fibonacci sequence
$$F_0= A+ B= 1$$
and
$$F_1=A\left(\frac{1+\sqrt{5}}{2}\right)+ B\left(\frac{1-\sqrt{5}}{2}\right)= 1$$
gives two equations to solve for A and B.

7. Aug 6, 2010

FeDeX_LaTeX

Sorry for the bump, but could you show me how you would solve for A and B?

I'm not able to solve simultaneous equations in this form;

A + B = 1
Ax + By = 1

Thanks.

8. Aug 6, 2010

Mentallic

Simply let B=1-A and then substitute this into the second equation, solve for A there and then substitute back into the first to find B.

9. Aug 7, 2010

FeDeX_LaTeX

Of course! Solving by substitution. Thanks, I forgot about doing that.

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