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 1.66667 2.66667 5 1.6 2.6 8 1.625 2.625 13 1.61538 2.61538 21 1.61905 2.61905 34 1.61765 2.61765 55 1.61818 2.61818 89 1.61798 2.61798 144 1.61806 2.61806 233 1.61803 2.61803 377 1.61804 2.61804 610 1.61803 2.61803 987 1.61803 2.61803 1597 1.61803 2.61803 2584 1.61803 2.61803 4181 1.61803 2.61803 6765 1.61803 2.61803

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