Fibonacci Ratios

The last three sections have done much to explain Golden geometry, but haven't explained how they relate to the topic of the Fibonacci series. Actually, the Golden Ratio and the Fibonacci numbers are two concepts that have a lot to do with each other.

The most important appearance of the Golden Ratio in Fibonacci mathematics is the Fibonacci ratio sequence. The Fibonacci ratio sequence is the series of numbers you get when you divide each Fibonacci number with the one that precedes it. Let's look at a few terms of this sequence.

 f(1) = 1 f(2) = 1 f(3) = 2 f(4) = 3 f(5) = 5 f(6) = 8 f(7) = 13 f(8) = 21
In order to create the first few terms of the sequence, we divide each Fibonacci number by the one before it. Since there is no Fibonacci number before f(1), we start with f(2). Dividing f(2) by f(1) gives a value of 1, so 1 is the first number of the Fibonacci ratio sequence, or r(1). Then we divide f(3) by f(2) to get the value of r(2), 2. Continuing in this fashion, we get:

r(1) = 1 / 1 = 1
r(2) = 2 / 1 = 2
r(3) = 3 / 2 = 1.5
r(4) = 5 / 3 = 1.67
r(5) = 8 / 5 = 1.6
r(6) = 13 / 8 = 1.625
r(7) = 21 / 13 = 1.615

 r(1) = 1 r(2) = 2 r(3) = 1.5 r(4) = 1.666666... r(5) = 1.6 r(6) = 1.625 r(7) = 1.6153846 r(8) = 1.6190476 r(9) = 1.6176471 r(10) = 1.6181818 r(11) = 1.6179775 r(12) = 1.6180555 r(13) = 1.6180258 r(14) = 1.6180371 r(15) = 1.6180328 First 15 terms

Do you notice a pattern beginning to emerge? We already proved that the decimal approximation of the Golden Ratio () is about 1.6180339. The farther we go in the Fibonacci ratio sequence, the closer we come to . However, since the real value of is an irrational number (can not be expressed as the ratio of two whole numbers), we can never really get exactly to in the sequence.

Note that the first, third, fifth, and seventh terms of the sequence are all less than , and the second, fourth, and sixth terms are all greater than . For a more visual demonstration of this relationship, see the In Action section.

Random Fibonacci number from the 1st through the 200th:

All contents, unless otherwise specified, are © 1999 by Matt Anderson, Jeffrey Frazier, and Kris Popendorf.
Created by Team 27890 for the 1999 competition.