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 *F*n denote the nth term, then the sequence can be specified by the recursive definition

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

Assume the limit exists, and call it “*x*”.

Since

,

we have

Thus

.

Now consider the continued fraction expression

.

If we let the value of this expression equal “*y*”, then we see,

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

NOTE:

is equivalent to

, which can be solved using the quadratic formula. Since we know *x* and *y* above are both positive, we get

, 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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The Departments of Mathematics and Computer Science is located on the third floor of the Behavioral and Social Sciences (BSS) Building Room 320. Telephone: (707)826-3143. Fax: (707)826-3140.

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