Real Analysis: Fibonnaci Numbers 1

[phyxius117]

Here is the problem:

Help please!!

 

[galactus]

This limit approaches the Golden Ratio. Which is about 1.618

It can be found if you scour the net.

 

[phyxius117]

Yes that is true but how would you go about proving that bn is between 1 and 2?

 

[galactus]

Knowing that \(\displaystyle F_{n+1}=F_{n}+F_{n-1}\), we can write:

\(\displaystyle \frac{F_{n}+F_{n-1}}{F_{n}}=1+\frac{F_{n-1}}{F_{n}}\)

There's a start. Since the Fibonacci sequence is an increasing sequence, the ratio \(\displaystyle \frac{F_{n-1}}{F_{n}}<1\)