[Guest]

Let F(n) denote the Fibonacci sequence, i.e.

F(n) = F(n-1) + F(n-2), F(0) = 1, F(1) = 1;

Find a closed form expression in terms of n for the ratio of consecutive numbers in this sequence: F(n)/F(n-1).

What is the limit of this ratio as n approaches infinity?

Edited October 18, 2009 by mmiguel1

[Guest]

the golden ratio: phi which is approx 1.61803398874

Edited October 18, 2009 by sgreene2

[Guest]

the golden ratio: phi which is approx 1.81803398874

Yes but you can represent this constant with integers and elementary functions to get an exact answer rather than an approximation.

[Guest]

phi happens to have the property

f - 1 = 1/f

You can use this equation to solve for an exact representation of phi

f^2 - f - 1 = o

(1 + (1 -- 4)^0.5)/2

(1 + sqrt(5))/2

Why phi has this property, I don't know, I'll try to figure that out

[Guest]

I think I remember finding this once a while ago. Let a and b be 2 consecutive terms in the sequence. The next number in the sequence must be a+b. Let us also assume that there is an exact ratio between any 2 consecutive terms in the sequence. Therefore:

b/a = (a+b)/b

b² - ab - a² = 0

solve for b: b = (a + a*sqrt(5))/2

Diving by a gives a value for a/b, or phi: phi = (1+sqrt5)/2

[Guest]

Cool.

You guys have found the answer to the second part of the question: "What is the limit of F(n)/F(n-1) as n approaches infinity?"

The first part is much harder and is basically equivalent to finding the closed form expression of F(n).

I will post a derivation later but give it a try.

[Guest]

I have written a detailed derivation of the answer to this problem. You can find it in .pdf format in the following link.

Answer

[bushindo]

I have written a detailed derivation of the answer to this problem. You can find it in .pdf format in the following link.

Answer

Nice solution. It's good to see another Latex user on this board.