Suppose a newlyborn pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was how many pairs will there be in one year? When attempting to solve this problem, a pattern is detected:
Figure 1: Recognizing the pattern of the "rabbit problem".
If we were to keep going month by month, the sequence formed would be 1,1,2,3,5,8,13,21 and so on. From here we notice that each new term is the sum of the previous two terms. The set of numbers is defined as the Fibonacci sequence. Mathematically speaking, this sequence is represented as:
The Fibonacci sequence has a plethora of applications in art and in nature. One frequent finding in nature involves the use of an even more powerful result of the Fibonacci sequence: phi and the golden ratio. The following is an example of what I will later discuss: the golden spiral.
Figure 2: The arrangement of the whorls on a pine cone follows a sequence of Fibonacci numbers.
The following example is just one of the numerous examples of the fascination applications found within the Fibonacci sequence in nature. Now, we turn to one of the most fundamental concepts of the Fibonacci sequence: the golden ratio.
Consider the ratio of the Fibonacci numbers (1,1,2,3,5,8, )
As, the sequence progresses, we notice that the sequence seems to converge and approach a number. The question is what exactly is that number? Answer:
One of the most interesting and frequent applications of phi is that of the golden rectangle. The golden rectangle is created in a way such that the area of the rectangle is phi (meaning that the length/width is one and the length/width is phi). Though ancient Greeks were...
...The Discovery of the Fibonacci Sequence
A man named Leonardo Pisano, who was known by his nickname, "Fibonacci", and named the series after himself, first discovered the Fibonacci sequence around 1200 A.D. The Fibonacci sequence is a sequence in which each term is the sum of the 2 numbers preceding it. The first 10 Fibonaccinumbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). Thesenumbers are obviously recursive.
Fibonacci was born around 1170 in Italy, and he died around 1240 in Italy, but the exact dates of his birth and death are not known. He played an important role in reviving ancient mathematics and made significant contributions of his own. Even though he was born in Italy, he was educated in North Africa where his father held a diplomatic post. He published a book called Liber abaci, in 1202, after his return to Italy and it was in this book that the Fibonaccinumbers were first discussed. It was based on bits of Arithmetic and Algebra that Fibonacci had accumulated during his travels with his father. Liber abaci introduced the HinduArabic placevalued decimal system and the use of Arabic numerals into Europe. Though people were interested, this book was somewhat controversial because it contradicted some of the foremost Roman and Grecian Mathematicians of the time, and even proved many...
...In order to calculate a certain term (number of months starting from January) the two previous terms must be known. These are then added together to give the desired month.
The table below shows the rabbit’s breeding numbers throughout the whole year.
The Mathematical recursive formula that represents this is:
Where: Tn= The desired month (January1, February2, March3, and so on) and where Tn>3
It can be clearly seen from the graph that the pattern/structure is exponential. This is due to the previous numbers being added in succession with the next, resulting in the ‘gap’ between each number to increase.
The trend in which the numbers follow is called a Fibonacci sequence and is often found in nature as well.
Many instances in which the Fibonacci Series is present in nature are that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonaccinumbers. However some plants such as the sneezewort plant (as seen left) can be seen demonstrating the Fibonacci pattern in succession. It happens on both the number of stems and number of leaves.
Another appearance of the Fibonacci Series in nature is that a lot of flowers and cone shaped structures have the...
...Fibonacci sequence in arithmetic sequence
The Fibonacci sequence is a series of numbers in which each number is the sum of the previous two. It starts with 0 and 1, which equals 1. Then 1 plus 2 equals 3, 2 plus 3 equals 5, and so on.
n mathematical terms, the sequence Fn of Fibonaccinumbers is defined by the recurrence relation
With seed values[1]
The Fibonaccinumbers are represented practically everywhere. In the petals on a flower, or the arrangement of leaves along a stem, you will find this sequence of numbers. The petals on most flowers display one of the Fibonaccinumbers. The numbers also appear in certain parts of sea shell formations. Parts of the human body also reveal these ratios, including the five fingers, and a thumb on each hand. Fibonacci also can be seen in a piano that produces harmony through a beautiful music. A piano has one keyboard with five black keys (sharps and flats) arranged in groups of two and three, and eight white keys (whole tones) for the 13 chromatic musical octaves.
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.
IN NATURE
A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.[52] This has the form
where n is the index...
...Anatolia College 
Mathematics HL investigation

The Fibonacci sequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonacci sequence can be defined as the following recursive function:
Fn=un1+ un2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0  F1  F2  F3  F4  F5  F6  F7 
0  1  1  2  3  5  8  13 
Segment 2: The Golden ratio
In order to define the golden ratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the golden ratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2x1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the formula of the golden ratio we discussed before. Therefore:
φ=x+1x...
...
Fibonaccinumber
From Wikipedia, the free encyclopedia
A tiling with squares whose side lengths are successive Fibonaccinumbers
An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.
In mathematics, the Fibonaccinumbers orFibonacci series or Fibonacci sequence are the numbers in the following integer sequence:[1][2]
0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; (sequence A000045 in OEIS)
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.
In mathematical terms, the sequence Fn of Fibonaccinumbers is defined by the recurrence relation
F_n = F_{n1} + F_{n2},\!\,
with seed values[3]
F_0 = 0,\; F_1 = 1.
The Fibonacci sequence is named after Leonardo Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics,[4] although the sequence had been described earlier in Indian mathematics.[5][6][7] By modern convention, the sequence begins either with F0 = 0 or with F1 = 1. The Liber Abaci began the sequence with F1 = 1, without an...
...
The Fibonacci sequence
The Fibonacci sequence is a series of numbers developed by Leonardo Fibonacci as a means of solving a practical problem. The original problem that Fibonacci investigated, in the year 1202, was about how fast rabbits could breed in ideal circumstances. Suppose a newly born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair, one male, one female, every month from the second month on. The question that Fibonacci posed was how many pairs will there be in one year?
At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
The Fibonacci sequence is the series of numbers, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… When squares are made with the widths, you get a nice spiral. If you look closely at the center of a daisy,...
...Introduction: The Fibonacci Series
The Fibonacci Series is a sequence of numbers first created by Leonardo Fibonacci (fibonachee) in 1202. It is a deceptively simple series, but its ramifications and
applications are nearly limitless. It has
fascinated and perplexed mathematicians
for over 700 years, and nearly everyone
who has worked with it has added a new
piece to the Fibonacci puzzle, a new tidbit
of information about the series and how it
works. Fibonacci mathematics is a
constantly expanding branch of number
theory, with more and more people being
Yellow flower with 8 petals, a Fibonacci
drawn into the complex subtleties of
Number.
Fibonacci's legacy.
The first two numbers in the series are one and one. To obtain each number of the
series, you simply add the two numbers that came before it. In other words, each number
of the series is the sum of the two numbers preceding it.
Note: Historically, some mathematicians have considered zero to be a Fibonaccinumber, placing it before the first 1 in the series. It is known as the zeroth Fibonaccinumber, and has no real practical merit. We will not consider zero to be a Fibonaccinumber in our discussion of the series....