Not sure what this Problem would fall under.

[shooterman]

I think there is an easier way to solve this and someone might tell me what this would fall under. The answer is 8/13.
"The objective is to find the exact value and explain how you get there." So I ask is there a shortcut to this?

 

[galactus]

Actually, it is 13/8 = 1.625

This part of the continued fraction for the Golden Ratio. The limit of the Fibonacci numbers approaches the Golden Ratio.

\(\displaystyle \lim_{n\to \infty}\frac{F_{n}}{F_{n+1}}=\frac{\sqrt{5}+1}{2}\approx 1.618\)

Let \(\displaystyle x=1+\frac{1}{x}\)

Then, we get \(\displaystyle x^{2}-x-1=0\)

The roots of which are the Golden Ratio.

 

[shooterman]

\(\displaystyle \lim_{n\to \infty}\frac{F_{n}}{F_{n+1}}=\frac{\sqrt{5}+1}{2}\approx 1.618\)


I don't get how you get the numbers for the limit of the Fibonacci numbers approaching the Golden Ratio. Yet I do get the Fibonacci sequence \(\displaystyle \frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{3}{5}\) then to 2+ don't see how it would make the 8/13. If you don't know what I mean I use this website for reference "http://www.friesian.com/golden.htm" Thx for the help.

 

[mmm4444bot]

Actually, it is 13/8 = 1.625

I do not know the changes made to the original post, but I get 8/13 right now.

I don't need a shortcut because it only takes 40 seconds to evaluate step-by-step.

 

[shooterman]

Re:

Actually, it is 13/8 = 1.625

I do not know the changes made to the original post, but I get 8/13 right now.

I don't need a shortcut because it only takes 40 seconds to evaluate step-by-step.

The change was a spelling error I put wave instead of way. *I guess by the time I finish asking for the shortcut I prob would have done the evaluate step-by-step anyway :lol: but the Fibonacci sequence intrigues me.