Question

What is the relationship between the Golden Ratio and Fibonacci sequence?

Answer 1

@Mertsj

Answer 2

\[\text{the ratio of the consecutive terms for the fibonacci } \lim_{n \rightarrow \infty}\frac{ a_{n+1} }{ a_n }=R\]
R is the golden ratio

Answer 3

You might want to look up Binet's Formula for that, more precisely Binet & De Moivre's formula.

Answer 4

\[\huge \phi=\frac{ 1+\sqrt{5} }{ 2}\]

Answer 5

\[F_n=\frac{ \phi^n-(-\phi)^{ -n} }{ \sqrt{5} }\]
Binet.

Answer 6

Or if you want to do it via induction, keep an eye on the parameters as you add successful exponents:

\[\phi^2-\phi-1=0\]
thus
\[\phi^2=\phi+1\]
and now multiplying both sides by phi.
\[\phi^3=\phi^2\phi=2\phi+1\]
\[\phi^4=\phi^3\phi=3\phi+2\]
and
\[\phi^5=\phi^4\phi=5\phi+3\]
The coefficients form the Fibonacci Sequence.

Answer 7

Thankcs guys :)