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- Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier
- Coconino Community College
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A famous and important sequence is the Fibonacci sequence, named after the Italian mathematician known as Leonardo Pisano, whose nickname was Fibonacci, and who lived from 1170 to 1230. This sequence is:
\[\{1,1,2,3,5,8,13,21,34,55, \ldots \ldots \ldots\} \nonumber \]
| This sequence is defined recursively. This means each term is defined by the previous terms. |
and so on.
The Fibonacci sequence is defined by  , for all  , when  and  . |
In other words, to get the next term in the sequence, add the two previous terms.
\[\{1,1,2,3,5,8,13,21,34,55,55+34=89,89+55=144, \cdots\} \nonumber \]
The notation that we will use to represent the Fibonacci sequence is as follows:
\[f_{1}=1, f_{2}=1, f_{3}=2, f_{4}=3, f_{5}=5, f_{6}=8, f_{7}=13, f_{8}=21, f_{9}=34, f_{10}=55, f_{11}=89, f_{12}=144, \ldots \nonumber \]
Example \(\PageIndex{1}\): Finding Fibonacci Numbers Recursively
Find the 13th, 14th, and 15th Fibonacci numbers using the above recursive definition for the Fibonacci sequence.
First, notice that there are already 12 Fibonacci numbers listed above, so to find the next three Fibonacci numbers, we simply add the two previous terms to get the next term as the definition states.
Therefore, the 13th, 14th, and 15th Fibonacci numbers are 233, 377, and 610 respectively.
Calculating terms of the Fibonacci sequence can be tedious when using the recursive formula, especially when finding terms with a large n. Luckily, a mathematician named Leonhard Euler discovered a formula for calculating any Fibonacci number. This formula was lost for about 100 years and was rediscovered by another mathematician named Jacques Binet. The original formula, known as Binet’s formula, is below.
| Binet’s Formula: The nth Fibonacci number is given by the following formula: \[f_{n}=\frac{\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right]}{\sqrt{5}} \nonumber \] |
| Binet’s formula is an example of an explicitly defined sequence. This means that terms of the sequence are not dependent on previous terms. |
A somewhat more user-friendly, simplified version of Binet’s formula is sometimes used instead of the one above.
| Binet’s Simplified Formula: The nth Fibonacci number is given by the following formula:
Note: The symbol |
means “round to the nearest integer.”Example \(\PageIndex{2}\): Finding 
Explicitly
Find the value of using Binet’s simplified formula.
Example \(\PageIndex{3}\): Finding 
Explicitly
Find the value of 
using Binet’s simplified formula.
Example \(\PageIndex{4}\): Finding 
Explicitly
Find the value of 
using Binet’s simplified formula.
All around us we can find the Fibonacci numbers in nature. The number of branches on some trees or the number of petals of some daisies are often Fibonacci numbers
Figure \(\PageIndex{4}\): Fibonacci Numbers and Daisies
a. Daisy with 13 petals b. Daisy with 21 petals
a. 
b. 
(Daisies, n.d.)
Fibonacci numbers also appear in spiral growth patterns such as the number of spirals on a cactus or in sunflowers seed beds.
Figure \(\PageIndex{5}\): Fibonacci Numbers and Spiral Growth
a. Cactus with 13 clockwise spirals b. Sunflower with 34 clockwise spirals and 55 counterclockwise spirals
a. 
b. 
(Cactus, n.d.) (Sunflower, n.d.)
Another interesting fact arises when looking at the ratios of consecutive Fibonacci numbers.
It appears that these ratios are approaching a number. The number that these ratios are getting closer to is a special number called the Golden Ratio which is denoted by 
(the Greek letter phi). You have seen this number in Binet’s formula.
| The Golden Ratio: \[\phi=\frac{1+\sqrt{5}}{2} \nonumber \] The Golden Ratio has the decimal approximation of \(\phi=1.6180339887\). |
The Golden Ratio is a special number for a variety of reasons. It is also called the divine proportion and it appears in art and architecture. It is claimed by some to be the most pleasing ratio to the eye. To find this ratio, the Greeks cut a length into two parts, and let the smaller piece equal one unit. The most pleasing cut is when the ratio of the whole length 
to the long piece 
is the same as the ratio of the long piece to the short piece 1.
1
cross-multiply to get
rearrange to get
solve this quadratic equation using the quadratic formula.
The Golden Ratio is a solution to the quadratic equation 
meaning it has the property 
. This means that if you want to square the Golden Ratio, just add one to it. To check this, just plug in 
.
It worked!
Another interesting relationship between the Golden Ratio and the Fibonacci sequence occurs when taking powers of 
.
And so on.
Notice that the coefficients of 
and the numbers added to the term are Fibonacci numbers. This can be generalized to a formula known as the Golden Power Rule.
| Golden Power Rule: \(\phi^{n}=f_{n} \phi+f_{n-1}\) where\(f_{n}\) is the nth Fibonacci number and \(\phi\) is the Golden Ratio. |
Example \(\PageIndex{5}\): Powers of the Golden Ratio
Find the following using the golden power rule: a. 
and b. 
