Fibonacci, also known as the Leonardo of Pisa, born in the early 1770’s AD in Pisa, Italy, has had a huge impact on today’s math, and is used in everyday jobs all over the world. After living with his dad, a North African educator, he discovered these ways of math by traveling along the Mediterranean Coast learning their ways of math. With the inspiration from the “HinduArabic” numerical system, Fibonacci created the 09 number system we still use to this day.
One of his most important and interesting discoveries is probably what is known as the Fibonacci sequence. It goes like this: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. He discovered this sequence through an experiment on an over population and breeding of rabbits. He then realized that if you add the last two numbers together you get the next one.
The Fibonacci sequence can be found almost anywhere including: architecture, economics, music, aesthetics, and most famously known, nature. For example the way seeds are arranged on a sunflower or pinecone, uses the Fibonacci sequence to prevent over crowding. It can also be seen in spiral galaxies, shells, the way water falls on a spider web, and even in your own body. Did you know that if you go from the bone in the tip of your finger to it’s middle it should be two fingernails long, followed by the base at about 5 fingernails, and the final bone goes all the way to about the middle of your palm which is the length of about 8 fingernails? There are other example of this in your body to such as a DNA strand is 34 by 21 angstroms. Mozart uses it in his worldknown sonatas by how many measures he puts in each section of his music. Or on a piano, if you look at the scale, there are 13 keys, 8 are white, and 5 black, which are split into groups of 2 and 3. When it comes to architecture, it’s been used as early as 2,560 BC on the Great Giza Pyramids. Leonardo DaVinci always tried to use this sequence throughout his artwork to,...
...The Discovery of the FibonacciSequence
A man named Leonardo Pisano, who was known by his nickname, "Fibonacci", and named the series after himself, first discovered the Fibonaccisequence around 1200 A.D. The Fibonaccisequence is a sequence in which each term is the sum of the 2 numbers preceding it. The first 10 Fibonacci numbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). These numbers 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 Fibonacci numbers 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...
...By: Rachel
Email: shed30rar@aol.com
Leonardo Fibonacci Leonardo Fibonacci was born in Pisa, Italy around 1175 to Guilielmo Bonacci. Leonardo's father was the secretary of the Republic of Pisa and directed the Pisan trading colony. His father intended on Leonardo becoming a merchant. His father enlisted him in the Pisan Republic, sending him to various countries. As Leonardo continued to travel with his father, he acquired mathematical skills while in Bugia. Fibonacci continued to study throughout his travels, which ended around the year 1200. Leonardo began writing books on number theory, practical problems of business mathematics, surveying, advanced problems in algebra and recreational mathematics. Leonardo's recreational problems became known as story problems and became mental challenges in the 13th century. Of all the books he wrote we still have copies of Liber abbaci (1202), Practica geometriae (1220), Flos (1225), and Liber Quadratorum. Sadly his books on commercial arithmetic Di minor guisa is lost as well as his commentary on Book X Euclid's Elements. One of Leonardo's contributions to mathematics was his introducing the Decimal Number system into Europe. He was one of the first people to introduce the HinduArabic number system into Europe. Fibonacci also introduced the Decimal Positional System, which originated from India and Arabia. Fibonacci wrote story problems in his book, Liber...
...Anatolia College 
Mathematics HL investigation

The Fibonaccisequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonaccisequence 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 Fibonaccisequence
The Fibonaccisequence 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...
...Fibonacci's Rabbits
The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.
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 dieand 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?
1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. 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 number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Another view of the Rabbit's Family Tree:
  
Both diagrams above represent the same information. Rabbits have been numbered to enable comparisons and to count them, as follows:
* All the rabbits born in the same month are of the same generation and are...
...SEQUENCE
* In mathematics, informally speaking, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
* For example, {M, A, R, Y} is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}. Also, the sequence {1, 1, 2, 3, 5, 8}, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers {2, 4, 6,...}. Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence { } is included in most notions of sequence, but may be excluded depending on the context.
ARITHMETIC SEQUENCE
* A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term...
...Arithmetic Sequence is a sequence of numbers in which each succeeding term differs from the preceding term by the same amount. This amount is known as the common difference and can be found using a specific formula by substituting the numbers from the word problem into the equation. When you plug in all the information, you are able to find out the money that needs to be spent and saved in the following word problems.
35. A person hired a firm to build a CB radio tower. The firm charges $100 for labor for the first 10 feet. After that, the cost of the labor for each succeeding 10 feet is $125 more than the preceding 10 feet will cost $125, the next ten feet will cost $150 etc. How much will it cost to build a 90 foot tower?
an=a1+ (n1) d
a125=100+ (1251) (150)
a125=100+124(150)
a125=100+18600
a125=18700
sn =n (a1 + an) / 2
= 125 (100+18700) /2
=125(1880) /2
=62.5 (18800) =1175000
The cost to build a 90foot tower is $11,750.
37. A person deposited $500 in a savings account that pays 5% annual interest that is compound yearly. At the end of 10 years, how much money will be in the savings account?
S+ (0.5) S n=10
S+ (1+0.5) r=1.05
S (1.05) a1= 500(1.05) =525
an= a1(rn1)
a10=525(1.059)
a10=525(1.551328216)
a10=814.4473134
The balance in the savings account at the end of 10 years will be $814.44.
I chose to use the Arithmetic...
...
This work MAT 126 Week 1 Assignment  Geometric and Arithmetic Sequence shows "Survey of Mathematical Methods" and contains solutions on the following problems:
First Problem: question 35 page 230
Second Problem: question 37 page 230
Mathematics  General Mathematics
Week One Written Assignment
Following completion of your readings, complete exercises 35 and 37 in the “Real World Applications” section on page 280 of Mathematics in Our World .
For each exercise, specify whether it involves an arithmetic sequence or a geometric sequence and use the proper formulas where applicable . Format your math work as shown in the Week One Assignment Guide and be concise in your reasoning. Plan the logic necessary to complete the exercise before you begin writing. For an example of the math required for this assignment, please review the Week One Assignment Guide .
The assignment must include ( a ) all math work required to answer the problems as well as ( b ) introduction and conclusion paragraphs.
Your introduction should include three to five sentences of general information about the topic at hand.
The body must contain a restatement of the problems and all math work, including the steps and formulas used to solve the problems.
Your conclusion must comprise a summary of the problems and the reason you selected a particular method to solve them. It would also be appropriate to...
...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 Fibonaccisequence. Mathematically speaking, this sequence is represented as:
The Fibonaccisequence 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 Fibonaccisequence: 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 Fibonaccisequence in nature. Now, we turn to one of the most fundamental concepts of the Fibonaccisequence: the golden ratio.
Consider the ratio of the...