Fibonnacci sequence

• September 14th 2012, 02:36 AM
mastermin346
Fibonnacci sequence
Can anyone explain if the Fibonnacci sequence can be placed into a parabolic or hyperbolic function?
• September 14th 2012, 06:29 PM
johnsomeone
Re: Fibonnacci sequence
If you're asking if there's a function $\phi:\mathbb{R}^+ \rightarrow \mathbb{R}$ such that the graph of $\phi$ is a parabola or hyperbola,

and that $\phi$ satistfies, for all positive integers $n, \ \phi(n) = F_n$, the $n^{th}$ Fibonnacci number, then the answer is no.

Parabolas and hyperbolas only have a few "free parameters", and the requirement $\phi(n) = F_n \forall n \in \mathbb{Z}^+$ would quickly exhaust them.

It would be possible that it algebraically worked out, but it doesn't. I'll do the parabola case, and leave to you the hyperbola ( $\phi(x) = b\sqrt{1+(x/a)^2}$).

If a parabola is the graph of a function $\phi$ that satisfies $\phi(n) = F_n$, then let $\phi(x) = ax^2 + bx + c$ for some $a, b, c \in \mathbb{R}$.

Then $\forall n \in \mathbb{Z}^+, \phi(n+2) = \phi(n+1) + \phi(n)$, so $a(n+2)^2 + b(n+2) + c = ( a(n+1)^2 + b (n+1) + c ) + ( an^2 + bn + c )$.

Thus $\forall n \in \mathbb{Z}^+, a $(n+2)^2 - (n+1)^2 - n^2$ + b$(n+2) - (n+1) - (n)$ + $c - c - c$ = 0$, so

$a(- n^2 -2n + 3 )+ b(-n + 1) - c = 0$, so $(-a)n^2 + (-2a-b)n + (3a+b-c) = 0$

The only way that could be true for all n is if each of those n-coefficients was 0. Thus $-a=0, -2a-b = 0, 3a+b-c = 0$.

That system of equations (3 equations in the 3 unknowns $a, b$, and $c$) has only one solution: $a = b = c = 0$.

That means the only quadratic that has a prayer of working is the 0 quadratic $\phi(x) = 0x^2+0x+0 = 0$.

But that obviously doesn't satisfy $\phi(n) = F_n \forall n \in \mathbb{Z}^+$, and it was the only one that had a chance.

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I suppose I should note that even if you consider rotated parabolas/hyperbolas, you're still going to quickly run out of free parameters.
(The standard equation for the graph of a generic conic section in the plane has 6 free parameters. That's more than the 3 a quadratic
polynomial has, but it's still less than the infinity required.).