The Fibonacci Sequence is a sequence generated by taking f(0) = 1, f(1) = 1, and f(n) = f(n - 2) + f(n - 1). Here are the first few terms in the Fibonacci Sequence:
1, 1, 2, 3, 5, 8, 13, 21, ...
More terms can be found in a spreadsheet at the bottom of the page.
There are many things that make the Fibonacci Sequence special, particularly its appearance in nature. However, this write up is on the mathematics of the Fibonacci Sequence. For more information on its appearance in nature, go to the following link:
http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
Another interesting fact is that when we take all of the ratios of consecutive terms in the Fibonacci Sequence, we find that they tend to a particular number. Here are a few such ratios:
As can be observed from these ratios, they tend to the same number, taking only the terms with odd values of n for their numerators approach this number from above (2, 1.667, 1.625, ...), and taking only the terms with even values of n for their numerator approaches this number from below (1, 1.5, 1.6, 1.619, ...).
When we look at some more terms (as can be seen in the spreadsheet at the bottom of the page), we can find that the number these ratios tend to is approximately 1.61003399.
This number is in fact, the golden ratio. The golden ratio can be described in many ways. One of these ways is a way to divide a line segment into two parts such that:
Here is a GSP file that shows the division of a line segment into the golden ratio: Golden Ratio
Dividing a line segment of length 2 into the golden ratio gives us a longer division of length 
- 1, and a shorter division of length 3 - . We can use these to define the golden ratio as:
, which also approximates to 1.61003399.
This does not prove mathematically that the ratios of consecutive terms in the Fibonacci Sequence tend to the golden ratio, but it does give a pretty strong argument for the case.
Similarly, we can take the ratios of every other term in the Fibonacci Sequence. As can be seen from the spreadsheet at the bottom of the page, these ratios tend to 2.61803399. When we square our value for the golden ratio, we get this same number:
= 2.61803399.
Assuming that ratios of consecutive terms in the Fibonacci Sequence do tend to the golden ratio, it is easy to prove that ratios of every other term tend to the square of the golden ratio. Here is such a proof:
note GR = the golden ratio =
Similarly, we can prove that the ratios of every third term tend to the golden ratio cubed, the ratios of every fourth term tend to the golden ratio to the fourth power, and so on and so forth.
The Fibonacci Sequence is not the only sequence that obeys the rule f(n) = f(n - 2) + f(n - 1). For instance, we can take the sequence of Lucas numbers, which also obeys this rule, and f(0) = 1, and f(1) = 3. Here are a the first few terms in the sequence of Lucas numbers:
1, 3, 4, 7, 11, 18, 29, 47, ...
More terms can be found in the spreadsheet at the bottom of the page.
Similarly to before, we can take ratios of consecutive Lucas numbers to find out if they tend to a ratio as well. Here are the first few such cases:
As can be viewed from the spreadsheet at the bottom of the page, consecutive ratios of these numbers also tend to 0.61803399. This time, the terms with odd values of n for their numerators approach this number from above (3, 1.75, 1.636, 1.621, ...), and terms with even values of n for their numerators approach this number from below (1.333, 1.571, 1.611, ...).
This leads us to the conjecture that no matter what the f(0) and f(1) are equal to, ratios of consecutive terms tend to the golden ratio. Here is what such an abstract sequence would look like, such that g(0) = a, g(1) = b, and f(n) = the nth term of the Fibonacci Sequence:
This tells us that no matter what a and b are equal to, any sequence that follows the addition of the previous two terms condition can be written as g(n) = f(n-2)a + f(n-1)b, which means that ratios of consecutive terms of such a sequence are equal to:
If one could show that the limit of all such terms tends to the golden ratio as n approaches infinity, then one could prove that this fact is true for any sequence of the form f(n) = f(n - 2) + f(n - 1).
Here is the spreadsheet referenced in many of the explorations above:
term = the nth term in the sequence
fib seq = the value of the Fibonacci Sequence for n
ratio = the value of the ratio of the nth term in the Fibonacci Sequence over the n-1th term in the Fibonacci Sequence
ratio per 2 = the value of the ratio of the nth term in the Fibonacci Sequence over the n-2th term in the Fibonacci Sequence
ratio per 3 = the value of the ratio of the nth term in the Fibonacci Sequence over the n-1th term in the Fibonacci Sequence
lucas = the value of the nth Lucas number
lucas ratio = the value of the ratio of the nth term in the sequence of Lucas numbers over the n-1th term in the sequence of Lucas numbers
abstract = the nth term in the sequence such that f(0) = a, f(1) = b, and f(n) = f(n-2) + f(n-1)