By Nami Youn


Write-up  #12


Investigating Fibonnaci Sequence

Introduction

In this write-up, I will generate and explore a Fibonnaci sequence using a spreadsheet-EXCEL.


1. Constructing a Fibonnaci Sequence 
Generate a Fibonnaci sequence in the frist column using   f(0) = 1, f(1) = 1,


f(n) = f(n-1) + f(n-2)

1 1 2
1 2 3
2 1.5 2.5
3 1.666666667 2.666666667
5 1.6 2.6
8 1.625 2.625
13 1.61538462 2.61538462
21 1.619047619 2.619047619
34 1.617647059 2.617647059
55 1.618181818 2.618181818
89 1.617977528 2.617977528
144 1.618055556 2.618055556
233 1.618025751 2.618025751
377 1.618037135 2.618037135
610 1.618032787 2.618032787
987 1.618034448 2.618034448
1597 1.618033813 2.618033813
2584 1.618034056 2.618034056
4181 1.618033963 2.618033963
6765 1.618033999 2.618033999

The frist column contains the numbers in the Fibonnaci sequence. the second column contains the ratio of each pair of adjacant terms in the sequence. As the number of terms increases, the ratio is almost equal to 1.6180339. This number is the Golden Ratio.
We can notice that the limit of the ratio of every second term is 1+ Golden ratio.
Click here to see an Excel worksheet.
 

Continuing this process, we can get the ratio of every third term, the fourth term.
Click here to see an Excel worksheet.


2.  Exploring sequences where f(0) and f(1) are some arbitrary integers other than 1.

a.  f(0) = 4, f(1) =  8,      f(n) = f(n-1) + f(n-2)

b.  f(0) = 5, f(1) =  3,      f(n) = f(n-1) + f(n-2)

c.  f(0) = 1, f(1) =  3,      f(n) = f(n-1) + f(n-2)

Click here to see an Excel worksheet.


3. A Lucas sequence---the case 2-(c)

   A Lucas sequence :  f(0) = 1, f(1) = 3,

f(n) = f(n-1) + f(n-2)





Click here to see an Excel worksheet 2

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