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 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 of x to the formula of the golden ratio we discussed before. Therefore: φ=x+1x φ=1+52+11+52 φ=1+52
Thus φ2=1+522 φ2=3+52 and F2φ+F1=1+52+1=3+52
Therefore:
φ2=F2φ+F1 We can simplify other powers of φ the same way, thus:
φ3=2+5 and φ4=35+72
In order to from a conjecture connecting φn, Fn and Fn1 we can apply the relationship we found for f2 to the other powers of f: F3φ+F2= 2+5 and F4φ+F3=35+72
By examining the last two relationships we can deduce that: φn=Fnφ+Fn1
We can prove the...
...
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...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...
...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,...
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used because it is the first letter of the name of the celebrated Greek sculptor Phidias (490430 BCE). He produced the famous statue of Zeus in the...
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In mathematics, two quantities are in the goldenratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b,
Where the Greek letter phi (φ) represents the goldenratio. Its value is:
The...
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