Assignment 12
Fibonacci Numbers and the Golden Ratio

by: Kelli Nipper

The Fibonacci numbers can be found in Pascal's triangle by drawing diagonals beginning at each 1 on the left side (passing under the 1 in the row above it). The numbers they pass through add up to the Fibonacci Sequence.

Pascal's Triangle
             



 1		              
           



 1		  



 1		            
         



 1		  



 2		  

 1		          
       



 1		  

 

 3

  


 3
  



 1		        
     

 

 1

  

 4
  



 6		  

 

 4

  


 1
      
   



 1		  



 5		  

 

 10

  

 10
  



 5		  



 1		    
 



 1		  

 

 6

  


 15
  



 20		  



 15		  



 6		  



 1		  

 

 1

  


 7
  



 21		  



 35		  



 35		  



 21		  



 7		  



 1

The following pattern is created:
1
1
1 + 1 = 2
1 + 2 = 3
1 + 3 + 1 = 5
1 + 4 + 3 = 8
1 + 5 + 6 + 1 = 13
1 + 6 + 10 = 21
(A few examples have been heighlighted above for clarity!)
Using Excel we can explore the Fibonacci sequence to find patterns.
The spreadsheet below compares ratios of different terms in the sequence as the number of terms increases:





 
Term Fibonacci Sequence Ratio of Adjacent Terms Ratio of 2nd term Ratio of 3rd term Ratio of 4th term
1 1
2 1 1
3 2 2 2
4 3 1.5 3 3
5 5 1.66666666666667 2.5 5 5
6 8 1.6 2.66666666666667 4 8
7 13 1.625 2.6 4.33333333333333 6.5
8 21 1.61538461538462 2.625 4.2 7
9 34 1.61904761904762 2.61538461538462 4.25 6.8
10 55 1.61764705882353 2.61904761904762 4.23076923076923 6.875

Click Here to view the complete spreadsheet for the Fibonacci Sequence.
Looking at the ratio of adjacent terms, we can quickly see that as n increases, the ration converges to 1.618033989. Consdering the ratio of every 2nd term of Fibonacci's sequence, as n increases, the ratio converges to 2.618033989 (which is one higher than the previous ratio.)
As n increases, the third and fourth ratios aproach 4.236067978 and 6.854101966 respectively.
Looking for a familiar relationship, I found that if you take the ratio of one ratio to the previous ratio, it also converges to the number 1.618033989.
Ratio of second term to ratio of first term

2.618033989 / 1.168033989 = 1.618033989

Ratio of third term to ratio of second term

4.236067978 / 2.618033989 = 1.618033989

Ratio of fourth term to ratio of third term

6.854101966 / 4.236067978 = 1.618033989

This value is an approximatation of the golden ratio.
Click Here for a discussion of how to find the exact value of the Golden ratio using the quadratic formula.
Click Here to see an investigation of the golden ratio.
Next, I explored the Lucas Sequence to find patterns in it.
The spreadsheet below compare ratios of different terms in the sequence as n increases.





 
Term Lucas Sequence Ratio of Adjacent Terms Ratio of 2nd term Ratio of 3rd term Ratio of 4th term
1 3
2 4 1.33333333333333
3 7 1.75 2.33333333333333
4 11 1.57142857142857 2.75 3.66666666666667
5 18 1.63636363636364 2.57142857142857 4.5 6
6 29 1.61111111111111 2.63636363636364 4.14285714285714 7.25
7 47 1.62068965517241 2.61111111111111 4.27272727272727 6.71428571428571
8 76 1.61702127659574 2.62068965517241 4.22222222222222 6.90909090909091
9 123 1.61842105263158 2.61702127659574 4.24137931034483 6.83333333333333
10 199 1.61788617886179 2.61842105263158 4.23404255319149 6.86206896551724

Click Here to view the complete spreadsheet for the Lucas Sequence.
For the Lucas Sequence, each of the ratios converge to exactly the same value as the ratios for the Fibonacci Sequence.
Click Here to learn about Lucas and Recursive Relations.
Finally, looking at an arbitrary sequence, we quickly see that the ratios converge to the same values as the ratios in the Fibonacci and Lucas sequences.
Spreadsheets are powerful tools in the classroom. Teachers can quickly demonstate ratios of terms and how rapidly numbers in a sequence (such as Fibonacci's) increase. By simply changing a seed value, the entire sequence changes.

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