Fibonnaci
Sequence
Rozina Essani
For this investigation I looked at the Fibonnaci sequence and applied it to Excel spreadsheets. I
have investigated the sequence for when n=0,1,2,É,47.
View the Excel workdsheets here.
Fibonnaci
Sequence
|
n |
f(n)=f(n-1)+f(n+2) |
f(n)/f(n+1) |
f(n)/f(n+2) |
f(n)/f(n+3) |
f(n)/f(n+4) |
|
0 |
1 |
|
|
|
|
|
1 |
1 |
1 |
|
|
|
|
2 |
2 |
0.5 |
0.5 |
|
|
|
3 |
3 |
0.666666667 |
0.333333333 |
0.333333333 |
|
|
4 |
5 |
0.6 |
0.4 |
0.2 |
0.2 |
|
5 |
8 |
0.625 |
0.375 |
0.25 |
0.125 |
.
.
.
|
42 |
433494437 |
0.618033989 |
0.381966011 |
0.236067977 |
0.145898034 |
|
43 |
701408733 |
0.618033989 |
0.381966011 |
0.236067977 |
0.145898034 |
|
44 |
1134903170 |
0.618033989 |
0.381966011 |
0.236067977 |
0.145898034 |
|
45 |
1836311903 |
0.618033989 |
0.381966011 |
0.236067977 |
0.145898034 |
|
46 |
2971215073 |
0.618033989 |
0.381966011 |
0.236067977 |
0.145898034 |
|
47 |
4807526976 |
0.618033989 |
0.381966011 |
0.236067977 |
0.145898034 |
The Fibonnaci sequence is defined
as f(n) = f(n-1) + f(n-2) where f(0)=1 and
f(1)=1. The sequence adds up to a number in billions when n reaches 47.
Let us first
investigate the ratio of each pair of adjacent terms in the Fibonnaci
sequence. We see that the ratio, f(n)/f(n+1) has a
limit which is 0.618033989. Now let us look at the ratio of every second term
in the Fibonnaci sequence. The first ratio of f(n)/f(n+2) is the second ratio of f(n)/f(n+1). This ratioÕs
limiting number is 0.381966011. Now let us look at the ratio of every third
term and every fourth term. Again the first entries for these ratios are the
second entries of the previous ratios respectively. The limit for f(n)/f(n+3) and f(n)/f(n+4) is 0.23606798 and 0.14589503
respectively.
When changing f(1)=1 to f(1)=3, the Lucas sequence, our ratios have the
same limits even though they do not began the same way as in the Fibonnaci sequence.
Lucas Sequence
|
n |
f(n)=f(n-1)+f(n+2) |
f(n)/f(n+1) |
f(n)/f(n+2) |
|
0 |
1 |
|
|
|
1 |
3 |
0.333333333 |
|
|
2 |
4 |
0.75 |
0.25 |
|
3 |
7 |
0.571428571 |
0.428571429 |
|
4 |
11 |
0.636363636 |
0.363636364 |
|
5 |
18 |
0.611111111 |
0.388888889 |
.
.
.
|
42 |
969323029 |
0.618033989 |
0.381966011 |
|
43 |
1568397607 |
0.618033989 |
0.381966011 |
|
44 |
2537720636 |
0.618033989 |
0.381966011 |
|
45 |
4106118243 |
0.618033989 |
0.381966011 |
|
46 |
6643838879 |
0.618033989 |
0.381966011 |
|
47 |
10749957122 |
0.618033989 |
0.381966011 |
Now if we take the
ratios of each pair of adjacent limits then we get a value of 1.61803399. This
in fact is the Golden Ratio.