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Old 04-14-2024, 02:15 AM
 
Location: Eugene, Oregon
1,413 posts, read 1,513,508 times
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(Note: the following question is just for personal interest. School and university were so many decades ago that it brings a tear to my eye to think of it. Definitely not asking anyone to do my homework.)

For anyone unfamiliar with the Fibonacci sequence, it's formed recursively starting with 0 and 1, and each member of the series thereafter is formed by adding the two preceding ones, i.e.:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,....

The Golden Ration is the solution for Φ to Φ^2 - Φ - 1 = 0, which works out to 1.6180339....

The interesting thing about this number is that the higher you go in the Fibonacci sequence, the ratio of any given number to the previous one comes nearer and nearer to the Golden Ratio.

So in the sequence above, 3/2 = 1.5 is obviously not Φ, but 13/5= 1.625 which is getting closer. And 89/55 1.6181818 which is very close.

My question is this: Is there a simple algebraic proof for this? Or any kind of proof? I mean, a proof that I don't need to have a Ph.D. in number theory to understand, because I don't.
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Old 04-18-2024, 10:03 AM
 
Location: Juneau, AK + Puna, HI
10,546 posts, read 7,739,679 times
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Proof for what, exactly? Phi is an irrational number, so the ratio of two real numbers will never exactly match it.
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Old 04-18-2024, 04:13 PM
 
Location: Sunnybrook Farm
4,511 posts, read 2,656,277 times
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The proposition would be, I guess,

for numbers in a fibonacci sequence, f(i) where F(i) = F(i-2) + F (i-1), lim (i --> infinity) F(i)/F(i-1) = PHI

I certainly don't have the tools to try to prove this.

It would be an interesting question to pose to a mathematics professor.

You've picked just one of the fibonacci sequences - if you start with a different number you get different numbers, of course - so a second question would be do all fibonacci sequences asymptote to PHI or just the one beginning with 1,1,2?
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Old 04-18-2024, 04:30 PM
 
Location: Victory Mansions, Airstrip One
6,750 posts, read 5,047,257 times
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Try this...

https://proofwiki.org/wiki/Ratio_of_...onacci_Numbers
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Old Yesterday, 05:35 AM
 
Location: Fortaleza, Northeast of Brazil
3,977 posts, read 6,783,287 times
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1 divided by Φ = Φ - 1

cool
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Old Today, 03:23 PM
 
Location: The Driftless Area, WI
7,242 posts, read 5,117,125 times
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Good question.

Trivial explanation-- some series are convergent (eg- Taylor or Maclaurin series or.Euler's number) That Fib series happens to converge on the value of the Golden Ratio.
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