Mathematics HL investigation
The Fibonacci sequence|
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
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 prove the...