Fibonacci Sequence and the Golden Ratio
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=un-1+ un-2
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=F1-F0F2=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 x2-x-1=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 Fn-1 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+Fn-1 We can
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=un-1+ un-2
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=F1-F0F2=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 x2-x-1=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 Fn-1 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+Fn-1 We can