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). These...
...Maths CRabbits Report
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...
...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 Fibonacci...
...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...
...
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 or...
...
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...
...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...
...computing large Fibonaccinumbers
Daisuke Takahashi
Department of Information and Computer Sciences, Saitama University, 255 ShimoOkubo, Urawashi, Saitama 3388570, Japan Received 13 March 2000; received in revised form 19 June 2000 Communicated by K. Iwama
Abstract We present a fast algorithm for computing large Fibonaccinumbers. It is known that the product of Lucas numbers algorithm uses the fewest bit operations...