Exploration 12 : Polar Equations
By: Taylor Adams
The Fibonacci sequence is represented by the formula
Given that f(0)=1 and f(1)=1.† Therefore, f(2) = f(1)+f(0) = 1+1 = 2.†
Using an Excel sheet, I generated the Fibonacci sequence in column A up to f(100).†† In column B, I constructed a ratio, of the terms in the Fibonacci sequence.† The ratio begins with f(1) and gives us † At f(2), we have 2.†
The values in this column fluctuate between 1 and 2 and look like they begin to approach some number.† By f(17), the ratio has approached 1.618034.† The ratio of every value after this point is essentially this number.†
It is interesting that the ratio of the Fibonacci numbers approach this number, because this is the golden ratio.† The golden ratio is represented by the formula
If we set x=, we get
If we look at the ratio of , as seen in column B of the Excel sheet, the ratio at f(n) holds this relationship with the ratio before it.† For example, the ratio of f(4)=1.5.
†This is also equivalent to
Letís see if this relationship carries with x2 and the ratio † To do this, letís first multiply through by x to get
This equation tells us that the ratio that the ratio †approaches should be the golden ratio squared, which is also equal to one added to the golden ratio.
If we look at the ratio of , as shown in column C of the Excel sheet, we can see that this relationship is true.† The values, to begin with, fluctuate between 2 and 3, but end up approaching a limit at 2.618034 by f(18).
We can visually see this ratio reaching the limit 2.618034 through the following graph:
When the golden ratio, x, is cubed, we get the equationÖ
This tells us that the values for the ratio †should approach.
The ratio †is shown in column D of the Excel sheet.† We can see that the values for this ratio do approach 4.236068.† In fact, they reach this limit at f(20).
Letís see of x4 is also the ratio of .† First, letís multiply x3 by x to getÖ
This tells us that the ratio of †should approach
If we look at this ratio shown in column E of the Excel sheet, we can see that this ratio does approach 6.854102 at f(22).
This should also be true for the ratio .
This tells us that the ratio †should approach
If we look at column F, we can see that this ratio does approach 11.09017 at f(21).
Letís see if this relationship occurs in another sequence.† The Lucas sequence, for example, is very similar to the Fibonacci sequence in that
Except f(0)=1 and f(1)=3.
I constructed the Lucas sequence in column H and set up the same ratios as I did for the Fibonacci sequence in columns I-M respectively.†
It is interesting to see that not only does this sequence approach some value for each ratio, but it also approaches the golden ratio and holds the same relationships when the golden ratio is raised by some power.† Actually, all sequences like these approach the golden ratio and have the same limits for the other ratios as well.