Solutions to our Post Card

Problem No. 2

 

Fibonacci card

 

If you cannot read the solution below, click on this link for a PDF version:
http://www.humboldt.edu/math/solutions/solution2.pdf

The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … is the famous Fibonacci sequence.  The first two terms in the sequence are 1. From then on, any other term in the sequence is the sum of the two preceding terms.  If we let Fn denote the nth term, then the sequence can be specified by the recursive definition
F sub 1 = 1, f sub 2 = 2
Fsub n plus 1 = F sub n plus F sub n minus 1 n = 2,3,4, . . .

The Fibonacci sequence grows without bound but ratios of successive Fibonacci numbers converge.  The first several ratios are
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666…
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538…
To determine what the ratios are converging to we need to evaluate  The limit of F sub n plus 1 over F sub n as n approaches infinity. 
Assume the limit exists, and call it “x”. 
Since

F sub n plus 1 equals F sub n plus F sub n minus 1 ,

we have
x equals the limit of F sub n minus 1 over F sub n as n approaches infinity equals the limit of F sub n plus F sub n minus 1 over F sub n as n approaches infinity equals the limit of one plus F sub n minus 1 over F sub n as n approaches infinity equals the limit of 1 plus 1 over F sub n divided by F sub n minus 1 equals 1 plus 1 over x  
Thus

x equals one plus one over x. 

Now consider the continued fraction expression

One plus one over (one plus one over (one plus one over . . .  . 

If we let the value of this expression equal “y”, then we seey equals one plus one over y,

which is the same equation we got when working with ratios of Fibonacci numbers!
NOTE:

x equals 1 plus 1 over x

is equivalent to

x squared minus x minus 1 equals zero, which can be solved using the quadratic formula.  Since we know x and y above are both positive, we get

x equals y equals (1 plus the square root of 5) over 2, the so-called “golden ratio”.  See http://en.wikipedia.org/wiki/Golden_ratio for much more on this fascinating number. Or contact anyone in the HSU Mathematics Department. The department’s web page is http://www.humboldt.edu/math/       

 

 

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