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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...

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