Question To Test Your Logic

[DoesntMatter]

OK so 3 people posted a question of "logic" so now I will too. If you want to answer it, go for it in this thread

What exactly is this number?

[Sanctuary]

Not sure if I did this right but 1.61803398874989??

[Initials]

This is the solution. If you want to figure it out on your own, don't read this I guess.

I don't directly see how turn this into a function of n, so let's try going somewhere with it and seeing what happens. That seems like the best course of action.

2,1/2... 3/2,2/3... 5/3,3/5... 8/5,5/8... 13/8,8/13
They're two sets of values, so I'll separate the one's I care about.

2/1, 3/2, 5/3, 8,5, 13/8...

Well, the first pattern developing is...

F(n)/F(n-1)

So, basically, some function evaluated at some value n divided by the function evaluated at the value of n one unit prior.

I'm not going to bother going any further with my testing up there, since you can see it's the Fibonacci sequence, so there's no point trying to find a function of n that explains the Fibonacci sequence.

Let's then define F(n) to be the Fibonacci sequence.

Lim (F(n)/F(n-1)) as n goes to infinity (sorry, I don't know how to notate that on a computer) = the number I'm trying to find.

Since there's no known function of n that satisfies the Fibonacci sequence, I don't believe there's any way to do this other than cranking out numbers and trying to see a trend... Or is there?

Yay, there is! I thought of something.

As you take the limit as n goes to infinity, you would be approaching the value that, when put into the function 1+(1/c) (c being the constant we are trying to find), equals itself.

So, 1+1/c = c, c^2-c-1 = 0, (c-(1/2))^2-1/4-1 = 0, (c-(1/2))^2 = (5/4), c-1/2 = sqrt(5/4), c = sqrt(5/4)+(1/2), c = (1/2)(sqrt(5)+1)

c = (1/2)(sqrt(5)+1)

So... That's φ (phi). I've never solved this whole thing before, but I have solved for φ before using that last equation, which is how I know the number has a name.

There's your answer.

Ah, that was fun! Thank you very much.

Just to reiterate, the exact value is .5(sqrt(5)+1).

[DoesntMatter]

You're welcome Writing the fraction as 1 + 1/c = c was the trick, to give c = 1/2 + sqrt.(5)/2

[Initials]

Yeah. I kept even the irrelevant stuff in there because, for one, I wanted to show my thought process, and for another I found it interesting that I inadvertently proved the Fibonacci sequence thing up there also equals φ.

The way you said that bothers me though, for some reason. I don't know why.

[DoesntMatter]

The way I said what?

Yeah though, that was interesting to note that about the ratio of the Fibonacci numbers

[Initials]

What you said about 'writing blah blah blah was the trick'.

I might be wrong, but it felt like it had some kind of condescending note to it.

Maybe not, but it seemed like it. Again, I don't know why.

[DoesntMatter]

Everything I say is condescending to you. Your perception is accurate

I know my personality is grating and uncouth. I know I've overstayed my welcome here too

[Initials]

Well that's a very friendly thing to say, isn't it?

Doesn't really bother me though. You seem pretty average. I've met much nicer people and I've met much meaner people.

[RSK]

finally got this!!!!! knew there was something more to this

very nice! i'm going to use this one at my next study group

[Sanctuary]

WTF math!

I hate math and I have no idea what you just did but I got the same answer.

All I did was plug the equation into an excel spreadsheet. Took me a minute to do it.