Question

Define the Golden Ratio and the Fibonacci Sequence.

Discuss the relationship of the Golden Ratio to the Fibonacci Sequence. Clearly explain their relationship.

Discuss the work of at least three men or women with the Golden Ratio and the Fibonacci Sequence. How they did contribute to these areas?

Answer 1

I know fibonacci sequence is sum of previous two elements

Answer 2

The formula most people know for the Fibonacci sequence is :
\[F_{n+2} = F_{n+1}+F_n\]
but this doesnt really help if you need the 100th term, you would only be able to get it if you knew the previous 99 terms.
So the question comes down to, is there is colsed formula, some function of n, call it, F(n), that if i wanted the 100th term, it would be F(100)? the answer is yes, and it is:
\[F(n) = \frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n}-(\frac{1-\sqrt{5}}{2})^{n}\]
in which you should recognize the golden ratio.
If you need this formula derived, i can write out a proof, although it uses Linear Algebra (more specifically, being able to diagonalize a matrix). if you dont know about that topic, my proof would be useless.

Answer 3

oops, add one more parenthesis to the very end of that formula.