Fibonacci Sequence and the Golden Ratio
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.
By showing the conjecture is true for n=1 we assumed that it is true for n=k and then proved that n=k+1 is true. Therefore it follows by the method of induction that P(n) is true. Segment 5: Formula of Fn We can use the equations we derived before in order to find a formula for Fn: We can achieve that by solving the system of the two equations which gives us: Fn=-1? n-? n-1? n-? Conclusion In this investigation we studied the concept of the golden ratio and we managed to connect it to the Fibonacci series by forming different conjectures and then proving them.
Ultimately we derived a formula for any term of the Fibonacci function, Fn in correlation with the golden ratio , ? , and it is the following: Fn=-1? n-? n-1? n-? We could further expand this investigation by testing more analytically the relationship between Fibonacci sequence and the golden ratio. This relationship has many interesting concepts which vary from a simple division of a term of the sequence by its previous one giving ? to more complex ones such as testing the limits and sums of such functions e. g. And =>