University of Georgia
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Fibonacci and the Golden Ratio
The relationship between the Fibonacci Sequence and
the Golden Ratio is a surprising one. We have two seemingly unrelated
topics producing the same exact number. Considering that this
number (or Golden Ratio) is nonrational, the occurance is beyond
coincidence. It calls for futher examination...
The Golden Ratio = (sqrt(5) + 1)/2 or about 1.618
The Golden Ratio is, perhaps, best visually displayed
in the Golden Rectangle. This rectangle has the property that
its length is in Golen Ratio with its width. As a consequence,
we can divide this rectangle into a square and a smaller rectangle
that is similar to the first. Let the following GSP sketch illustrate:
The Fibonacci Sequence is one where each term is defined
as the sum of the two previous terms:
We can create this sequence easily in a spreadsheet,
using the formula above. This has been done in the center column
of the spreadsheet below:
1 
1 

2 
1 
1 
3 
2 
2 
4 
3 
1.5 
5 
5 
1.66666666666667 
6 
8 
1.6 
7 
13 
1.625 
8 
21 
1.61538461538462 
9 
34 
1.61904761904762 
10 
55 
1.61764705882353 
11 
89 
1.61818181818182 
12 
144 
1.61797752808989 
13 
233 
1.61805555555556 
14 
377 
1.61802575107296 
15 
610 
1.61803713527851 
16 
987 
1.61803278688525 
17 
1597 
1.61803444782168 
18 
2584 
1.61803381340013 
19 
4181 
1.61803405572755 
20 
6765 
1.61803396316671 
21 
10946 
1.6180339985218 
22 
17711 
1.61803398501736 
23 
28657 
1.6180339901756 
24 
46368 
1.61803398820533 
25 
75025 
1.6180339889579 
26 
121393 
1.61803398867044 
27 
196418 
1.61803398878024 
28 
317811 
1.6180339887383 
29 
514229 
1.61803398875432 
30 
832040 
1.6180339887482 
31 
1346269 
1.61803398875054 
32 
2178309 
1.61803398874965 
33 
3524578 
1.61803398874999 
34 
5702887 
1.61803398874986 
35 
9227465 
1.61803398874991 
36 
14930352 
1.61803398874989 
37 
24157817 
1.6180339887499 
38 
39088169 
1.61803398874989 
39 
63245986 
1.6180339887499 
40 
102334155 
1.61803398874989 
41 
165580141 
1.6180339887499 
42 
267914296 
1.6180339887499 
43 
433494437 
1.6180339887499 
44 
701408733 
1.6180339887499 
45 
1134903170 
1.6180339887499 
46 
1836311903 
1.6180339887499 
47 
2971215073 
1.6180339887499 
48 
4807526976 
1.6180339887499 
49 
7778742049 
1.6180339887499 
50 
12586269025 
1.6180339887499 
We have also taken the ratio of every two consecutive
terms, in the right column. If we take the limit of this ratio
as the terms get larger... the Golden Ratio!
Understanding the Relationship
Remember, now, that the Golden Rectangle can be divided
into a square and another Golden Rectangle. In fact, we can repeat
this process time and again. Suppose we start with a rectangle
with length Y and width X. When we divide this rectangle into
a square and a new Golden Rectangle, we get length X and width
YX for the new rectangle. Since the sides of the rectangles are
in the same proportion (namely 1.618...), we get:
Now, let's return to the Fibonacci Sequence. The ratio
of consecutive terms forms a sequence itself. Suppose we know
that this sequence converges to some real number (we can prove
this in a lesson on geometric sequences). Then, for large values
of n:
But , so
And, if we replace with Y and
with X, we have
The Golen Ratio!