This website uses cookies to ensure you have the best experience. Learn more

Fibonacci Sequence Introduction Essay

1534 words - 7 pages

FIBONACCI SEQUENCE Shafira Chairunnisa | 11 Blue
INTRODUCTION
Before we get into the details of Fibonacci sequence, let us go back to the basic definition of a sequence in Mathematics. According to The Free Dictionary by Farlex, a sequence is an ordered set of mathematical quantities called terms. There are two types of sequence: arithmetic and geometric sequence. An arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms.
1, 3, 5, 7, 9,….
The numbers in the sequence are called terms. Thus, 1 is the first term, 3 is the second term, 5 is the third term, and so forth. The symbol Un denotes the first term of a ...view middle of the document...

So now we raise up the problems. Having known the definition of Fibonacci sequence, can we consider it as an arithmetic or geometric? If it falls into any of those two categories, can we determine its general formula? Is there any other way to describe Fibonacci sequence using symbols? And what are the examples of this sequence in nature?

BODY
The General Term for Fibbonaci sequence
Theoritically, Fibonacci sequence doesn’t fall in the category for arithmetic nor geometric sequence. The ratio of or the difference between successive terms is not constant. However, we still an find the general formula for this sequence, although we don’t use Un, but Fn for Fibonacci.
First of all, we have to look again at (1). From the sequence, we can see that:
5= 3+2
F6= F5+ F4
F5+1= F5+ F5-1
Therefore, we can conclude that
 ,     (2)
Now, we assume that the sequence Fn has the form   , where   is a real parameter.
We substitute  in to (2)
                                                                                                 (3)
or equivalent                 
Since ,  , thus the last equation becomes  
Using quadratic formula for factorisation ( ), we’ll get:
  and               
Thus,    , . This verifies (1). Therefore, we can draw a conclusion that (1) can have more solutions. There are finite number of sequences that verifies (1), and one of them is:
                           (4)
The C1 and C2 in here are fixed real numbers.
For n = 0 and n = 1 in (4), we obtain a set of simultaneous equations
                                              
having the solutions  
Substituting the value of C1 and C2, we can conclude that the general term for Fibonacci sequence is                           N.    
                       
                       
Expressing Fibonacci sequence using Golden Ratio
Golden ratio, denoted φ, is a number equal to 1.61803398874989484820… or approximately 1.618. Here’s a concept and idea of the golden ratio.

Let’s say you divide a line into two parts, one part is longer than the other. When you divide the longer part by the shorter part, the result will be equal to the length of the line divided by the longer part. Whatever value of a and b you put, the result must be approximately 1.618. This number is the golden ratio.
Let us take a look again at the Fibonacci sequence.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…
Then, we try to find the ratio of each successive terms.
A | B |   | B / A |
2 | 3 |   | 1.5 |
3 | 5 |   | 1.666666666... |
5 | 8 |   | 1.6 |
8 | 13 |   | 1.625 |
... | ... |   | ... |
144 | 233 |   | 1.618055556... |
233 | 377 |   | 1.618025751... |
... | ... |   | ... |
If we pay attention closely, those ratios are close to the golden ratio. The bigger the Fibonacci number, the closer the ratio of successive terms to the golden ratio.
With golden ratio, we can find the value of any Fibonacci number, by using this formuka:

Whatever value you put as...

Other Papers Like Fibonacci Sequence- Introduction

Rational Emotional Behavior Therapy Case Study Conceptualization And Treatment Plan

2140 words - 9 pages Rational Emotional Behavior Therapy Case Study of Sarah: A Conceptualization and Treatment Plan Rational emotive behavior therapy, REBT, was developed by Albert Ellis and holds the central belief that the events in our lives do not cause our disturbances but that they are instead caused by our view of the events (Murdock, 2009). Murdock (2009) states that “people are seen as responsible for their behavior” (p. 279) but, because they are

Holidays In Albania Essay

1636 words - 7 pages Have you ever thought about having exciting and incredibly cheap vacations? Albania might be the right choice. According to My Travel Guide, Albania is ranked the fourth among ten places worth visiting in Eastern Europe (“Top 10 Eastern European Destinations”). One can encounter three kinds of vacations in this Mediterranean country: winter, summer, and cultural. The ideal places to spend your winter vacations are the Albanian Alps. They are

A Heart Of Darkness

1748 words - 7 pages In this short story, there are frequent significant subject and ideas that make the story, "A Heart of Darkness," by Joseph Conrad, and haunting novel. The main theme is absolute white power over the natives. The theme validates the corruption, and the dependence caused by the white people as they took over the Congo. White men were giving all the power; they had no self-control, and in the end they did not use wisely. The white men became

The Aspects Of Vulnerability Among The Exploited In Medical Research

2287 words - 10 pages Essentially, everyone is in some state of vulnerability. However, some of us are more susceptible to harm due to our vulnerabilities. The susceptible are the individuals with the greatest risk. These individuals risk the loss of their autonomy, and maybe even their lives. Vulnerable populations can be found in every subset of society. However, as previously mentioned, there are some vulnerable populations that are at an even greater risk than

The Hitchhiker’S Guide To The Galaxy

1171 words - 5 pages The Hitchhiker’s Guide to the Galaxy As the human race makes life-changing discoveries, it is made apparent that there is always more to learn as the universe, instead of becoming familiar, is becoming absurd. The Hitchhiker’s Guide to the Galaxy, written by Douglas Adams, as well as the 2005 film adaption, portrays absurdity to be an all-encompassing system in the universe. Through the introduction and attempt to understand lack of reason

The Ford Motor Company Wage Increase Of 1914 And The Theory Of Incentives And Efficiency Wages

1252 words - 6 pages ‘It’s not the employer who pays the wages. Employers only handle the money. It is the customer who pays the wages’ (Henry Ford, cited in Johnson and Weinstein 2004, p. 2). When the Ford Motor Company announced that it would more than double the wages of its workers in January 1914 to a ‘five-dollar day’ minimum, was this a contradiction to Henry Ford’s statement? If customers are actually the ultimate payers of wages, then more than

Historical Analysis Of The Economical Breakthroughs Of The Industrial Revolution

1396 words - 6 pages A Historical Analysis of the Economical Breakthroughs of the Industrial Revolution During the Industrial Revolution, many elements of society experienced huge breakthroughs that would change the way they functioned forever. Economics were definitely one of them. With many new inventions and many factories appearing, many, many, more resources were being created than ever before. Also, all of the revenue from these resources was being given to

Leadership Portrayed in Monologue from Shakespeare’s Henry V

1214 words - 5 pages Leadership is defined as a socially constructed process and which also affect organizational future outcomes. Leader is someone at high position who have overall duty for an organization, she or he decide what to do and the way how to achieve it. (Carter and Greer, 2013)The role of leader is extremely important for an organization, leader use their own power to influence the followers though many different ways such as motivation in order to

Cuba Civil Rights

1906 words - 8 pages Picture a country that has limits on how much you can make random government imprisonments and more what do you do. That is the situation for the eleven million seventy-five thousand two hundred forty-four people of Cuba since the takeover of Fidel Castro. The US is opposed to Castro taking control of Cuba. Castro managed to ease the United States nerves when he said that Cuba was against all forms of communism. According to Lana Wylie the Union

US Freight System

1273 words - 6 pages General Environment Each American requires the movement of approximately 40 tons of freight per year across the freight network. This includes everything from shirts to lawn mower to orange juice. • As the U.S. population expands, the U.S. freight system will be called upon to meet the demands of a larger population • Between 2010 and 2035, the system will experience a 22 percent increase in the total amount of tonnage it moves • By 2050, with

Democratic Peace Theory

1852 words - 8 pages Intro: US Presidents have made it a goal during their term(s) in office to establish a good relationship with foreign countries and even try to improve upon existing connections with our allies. Some believe it is to prevent conflicts between the countries while others dispute that it is a threat assessment by the United States to pick and choose their friends and enemies. Preventing conflict between two democracies or countries that practice

Related Essays

Fibonacci Essay

676 words - 3 pages . Fibonacci had passed away in 12402, but was still well known for his works as well as being recognizable for other achievements. His most famous achievement is the Fibonacci sequence as it’s considered to be his most important accomplishment to the introduction of Hindu-Arabic mathematics into western culture for being a compilation of known techniques for a new audience. He was also responsible for the introduction of Arab mathematics to

Gsl Scientific Library Essay

4796 words - 20 pages . A generalization of the generator to 32 bits is provided by gsl_rng_uni32. The original source code is available from NETLIB. [Generator] This is the SLATEC random number generator RAND. It is ancient. The original source code is available from NETLIB. [Generator] This is the ZUFALL lagged Fibonacci series generator of Peterson. Its sequence is, t = un−273 + un−607 un = t − floor(t) The original source code is available from NETLIB. For more

Freedom And Responsibility Essay

2141 words - 9 pages Built within the Constitution of the United States are specifically defined freedoms that are guaranteed to all citizens. Conversely, with every constitutional freedom there comes a corresponding responsibility. On September 25, 1789, the state legislature’s twelve proposed amendments were transmitted by congress, the first two dealing with congressional representation and congressional pay. The following numbers three through twelve were

Hate Crime Laws Essay

2348 words - 10 pages On June 7, 1998, 49-year-old James Byrd Jr. of Texas accepted a ride from three white men, who then beat him severely, urinated on him, chained him by his ankles to the back of their pick-up truck, dragged him for three miles into the countryside, and dumped his corpse in front of an African-American cemetery (Graczyk). A little over a year later, a jury sentenced ring leader John King to death by lethal injection (“Man Executed for Dragging