INTRODUCTION The Fibonacci Series is a sequence of numbers first created by Leonardo Fibonacci (fi-bo-na-chee) 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 drawn into the complex subtleties of 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 Fibonacci number, placing it before the first 1 in the series. It is known as the zeroth Fibonacci number, and has no real practical merit. We will not consider zero to be a Fibonacci number in our discussion of the series. THE SERIES An Introduction to the Series

The Fibonacci series is defined recursively. That is, in order to find each term of the series using the definition, you have to find all the terms that precede it. This makes finding the nth term very difficult for large values of n, as you must find every term that comes before. H owever, there could be a way to find Fibonacci numbers without using the definition. If this were possible, one would be able to find the nth term of the series simply by plugging n into a mathematical formula. In 1843, Jacques Philippe Marie Binet discovered just such a formula for finding the nth term of the Fibonacci series. The formula itself looks like this: BINET S FORMULA

THE SUCCESSOR FORMULA Suppose we know a term in the Fibonacci series and we want to know the term that directly follows it. How would we go about doing this? Without knowing the term that precedes the term we know, we can't use the definition of the series. We could use Binet's formula, but that takes a lot of work. What if there were an easier way to calculate a term of the series knowing only the term that precedes it? There is, of course, such a way, known as the Successor Formula, as it finds the successor to each term. This formula uses a function known as the "greatest integer" function. The greatest integer function is denoted [ x ] , and it is defined as the greatest integer less than or equal to x. For example: [ 4.2 ] = 4 [ 4 ] = 4 The Successor Formula takes the argument of any term of the Fibonacci series and is of the form: Note that the value of x in this equation is not the numerical order of the Fibonacci number, it is the Fibonacci number itself. For instance, if you put 3 into the Successor Formula, you will not get the 4th Fibonacci number, you will get the Fibonacci number that comes after 3 in the series.

BIONOMIAL FORM There is one more way that we know to calculate the nth term of the Fibonacci series; a very complicated one, in fact. This method is recommended to those already familiar with summation notation, binomial theory, and Pascal's Triangle. Every Fibonacci number can be expressed as the sum of a certain number of entries in Pascal's Triangle, as such: Note that the notation stands for "binomial n, k" and is the entry in the nth row and kth column of Pascal's Triangle. However, when finding rows and columns of Pascal's Triangle, one must be very careful to remember that the first row and column is always counted as 0. Therefore, the first row of Pascal's Triangle is actually the second one down, etc. Also, it is important to notice one other thing about binomial notation. If the lower number between the parentheses is larger than the upper number, the value of the term is always zero. That is, if k is greater than n, then "binomial n, k" is zero.

The Fibonacci Spiral is a geometric spiral whose growth is regulated by the Fibonacci Series. Its sudden, almost exponential growth parallels the rapid growth of the series itself. FIBONACCI SPIRAL

FIBBONACCI NUMBERS IN NATURE Fibonacci sequences have been noted to appear in biological settings,such as the branching patterns of leaves in grasses and flowers , branching in bushes and trees, the arrangement of pines on a pine cone , seeds on a raspberry , and spiral patterns in horns and shells (see phyllotaxis ). The scales on the surface of a pineapple are arranged in two interlocking spirals, eight spirals in one direction, thirteen in the other; each being a Fibonacci number. Przemyslaw Prusinkiewicz has advanced the idea that these can be in part understood as the expression of certain algebraic constraints on free groups , specifically as certain Lindenmayer grammars Generally one sees Fibonacci numbers arise in the study of the fractal Fuchsian groups and Kleinian groups , and systems that possess such symmetries. For example, the solutions to reaction-diffusion differential equations (such as that seen in the Belousov-Zhabotinsky reaction ) can show such a patterning; in biology, genes often express themselves through gene regulatory networks , that is, in terms of several enzymes controlling a reaction, which can be modelled with reaction-diffusion equations. Such systems rarely give the Fibonacci sequence exactly or directly; rather, the relationship occurs deeper in the theory. Similar patterns also occur in non-biological systems, such as in sphere packing models

IDENTITIES F ( n + 1) = F ( n ) + F ( n − 1) F (0) + F (1) + F (2) + … + F ( n ) = F ( n + 2) − 1 F (1) + 2 F (2) + 3 F (3) + … + n F ( n ) = n F ( n + 2) − F ( n + 3) + 2 These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here. In particular, F ( n ) can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F (0) = 0, meaning no sum will add up to −1, and that F (1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.

F ( n + 1) = F ( n ) + F ( n − 1)

F (0) + F (1) + F (2) + … + F ( n ) = F ( n + 2) − 1

F (1) + 2 F (2) + 3 F (3) + … + n F ( n ) = n F ( n + 2) − F ( n + 3) + 2

These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here. In particular, F ( n ) can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F (0) = 0, meaning no sum will add up to −1, and that F (1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.

The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers. Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation , which led to his original solution of Hilbert's tenth problem . The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle (see binomial coefficient ). Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem , and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. Fibonacci numbers are also used by some pseudorandom number generators . In music Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements. Examples include Béla Bartók 's Music for Strings, Percussion, and Celesta . In addition, the syllables of the lyrics of parts of the Tool song Lateralus follow the Fibonacci sequence in each line, for instance "Black/Then/White are/All I see/In my infancy/Red and yellow then came to be". Since the conversion factor 1.609 for miles to kilometers is close to the golden mean φ, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in base φ being shifted. To go from kilometers to miles shift the register down the Fibonacci sequence instead. APPLICATIONS

POPULAR CULTURE The Fibonacci sequence plays a small part in the bestselling novel and film The Da Vinci Code A part of the Fibonacci sequence is used as a code in Matthew Reilly's Ice Station . Tool 's song " Lateralus " from the album of the same name features the Fibonacci sequence symbolically in the verses of the song. The syllables in the first verse count 1, 1, 2, 3, 5, 8, 5, 3, 13, 8, 5, 3. Similarly, on Tool's 10,000 Days album there has already been speculation to more Fibonacci references embedded within the album. Marilyn Manson is another artist who has employed the Fibonacci sequence. He uses the sequence overtly in a watercolor painting entitled "Fibonacci" during his Holy Wood era, which it should be noted, uses bees as focal points. More discreetly, Manson used the sequence in the interior album art of Antichrist Superstar in his depiction of "The Vitruvian Man ", in the vein of Leonardo DaVinci 's work which was also based on the sequence. There is also speculation that some of the beats in the songs on the album Holy Wood (In the Shadow of the Valley of Death) are based on the Fibonacci sequence as well. Along with the concepts of the golden rectangle and golden spiral , the Fibonacci sequence is used in Darren Aronofsky 's indie film π (1998) Referenced in the film of The Phantom Tollbooth . Used in Steven Spielberg's miniseries Taken . It was also used as a key plot point in an episode of the Disney original television series So Weird . In a FoxTrot comic, Jason and Marcus are playing football. Jason yells, "Hut 0! Hut 1! Hut 1! Hut 2!" all the way until "Hut 13!" in the Fibonacci sequence. Marcus yells, "Is it the Fibonacci sequence?" Jason says, "Correct! Touchdown, Marcus!" BT ( Brian Transeau ) released a dance track in 2000, entitled the "Fibonacci Sequence," which features a sample of a reading of the sequence. Dr. Steel released a song titled "Fibonacci Sequence" in 2005.

PROGRAMMING OF FIBONACCI SERIES IN QBASIC CLS X=1 Y=1 FOR X=1 TO 100 ( THIS VALUE OF 100 CAN BE REPLACED BY A LARGER VALUE TO GET MORE OUTPUTS) Z=X+Y X=Y Y=Z PRINT Z+1 NEXT X