Mathematics
Education
EMAT 6680,
Professor Wilson
Exploration 12, the Fibonacci sequence by Ursula Kirk
Generate
a Fibonacci sequence in the first column using
f(0) = 1 and f(1) = 1
f(n) = f(n-1) + f(n-2)
a.
Construct the
ratio of each pair of adjacent terms in the Fibonacci sequence. What happens as
n increases? What about the ratio of every second term?
We can obtain the
Fibonacci sequence by using the recursive formula below when
We continue constructing
the sequence the same way; by the time we get to 
we have the sequence
as shown below. In this sequence, our numbers increase very rapidly as every
term is the sum of the two terms that came before.
|
f(0) |
0 |
|
f(1) |
1 |
|
f(2) |
1 |
|
f(3) |
2 |
|
f(4) |
3 |
|
f(5) |
5 |
|
f(6) |
8 |
|
f(7) |
13 |
|
f(8) |
21 |
|
f(9) |
34 |
|
f(10) |
55 |
|
f(11) |
89 |
|
f(12) |
144 |
|
f(13) |
233 |
|
f(14) |
377 |
|
f(15) |
610 |
|
f(16) |
987 |
|
f(17) |
1597 |
|
f(18) |
2584 |
|
f(19) |
4181 |
|
f(20) |
6765 |
Next, we explore the sequence that we obtain
from the ratio of the adjacent terms of the Fibonacci sequence.
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In this sequence, we
can see that as n gets larger, the ratio of each pair of adjacent terms of
the Fibonacci sequence converges to 1.61803399 which is the Golden Mean.
Next, we explore the sequence that we obtain from the ratio of
every second term of the Fibonacci sequence.
In this sequence, we
can see that as n gets larger, the ratio of every second terms of the
Fibonacci sequence converges to 2.61803399 which is the Golden Mean +1. b.
Explore
sequences where f(0) and f(1) are some arbitrary integers other than 1. If
f(0)=1 and f(1) = 3, then your sequence is a Lucas Sequence. All such
sequences, however, have the same limit of the ratio of successive terms. The arbitrary integers I choose are
Next, I will create two more sequences. One is
the ratio of each pair of adjacent terms in the Fibonacci sequence. The
second sequence is the ratio of every second term.
It is very interesting to observe that even though the values of my own “Fibonacci” sequence are different to the real Fibonacci sequence, the third column still approaches the Golden Mean, and the fourth column also approaches the Golden Mean +1.
The Lucas sequence has the values of
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