Question

The Fibonacci numbers exhibit an interesting property. The Fibonacci numbers obey the following sequence:
F0=1, F1=1, Fn=Fn-1+Fn-2
If a number is divided by the previous to it, you can get certain values as you go up the sequene, such that, for any Fibonnaci number Fn, it is true that the limit of Fn+1/Fn as n reaches infinity equals the Golden Ratio.
Describe how you can find the value of the Golden Ratio using this limit.

Answer 1

If you start with the definition
\[ F_{n+1} = F_n + F_{n-1} \]
divide both sides by \( F_n\)
\[ \frac{F_{n+1}}{F_n}=\frac{F_n }{F_n } + \frac{F_{n-1}}{F_n} \]

Answer 2

now take the limit
\[\lim_{n \rightarrow \infty}\left( \frac{F_{n+1}}{F_n}=\frac{F_n }{F_n } + \frac{F_{n-1}}{F_n} \right)\]

simplify Fn/Fn= 1. Also the limit is the limit of each term, so we get
\[\lim_{n \rightarrow \infty} \frac{F_{n+1}}{F_n}= 1 + \lim_{n \rightarrow \infty} \frac{F_{n-1}}{F_n} \\
\lim_{n \rightarrow \infty} \frac{F_{n+1}}{F_n}= 1 + \lim_{n \rightarrow \infty}\frac{1}{ \frac{F_n} {F_{n-1}}}\\
=\lim_{n \rightarrow \infty} \frac{F_{n+1}}{F_n}= 1 + \frac{1}{ \lim_{n \rightarrow \infty} \frac{F_n} {F_{n-1}}}
\]

we now replace the limit of the fibonacci sequence with the golden ratio, denoted by \( \phi\)
notice if we "flip" the last term, its limit is also \( \phi \)
we get
\[ \phi = 1 + \frac{1}{\phi} \]

Answer 3

That makes so much sense! Thank you!!!

Answer 4

the last step is solve for \(\phi\)

Answer 5

Can we assume the limit converges?

Answer 6

the question states
it is true that the limit of Fn+1/Fn as n reaches infinity equals the Golden Ratio.

so the (presumably very knowledgable) asker knows it converges.