The fibonacci sequence has intrigued mathematicians for centuries. It can be seen in nature, in the spirals of a pinecone or the arrangement of leaves on a stem. Leonardo pisanobigollo (1170-1250) discovered the sequence while he was trying to find an efficient solution to an accounting problem about rabbits.

The Fibonacci numbers are useful in several areas of mathematics and appear frequently in various fields including biology, physics, architecture, music theory, cryptography, etc. They are also used in many daily activities such as pricing problems and stock trading.

The fibonacci sequence consists of numbers that have values equal to the sum of values before it in the sequence. It is usually written with 0, 1, 1, 2, 3, 5, etc.

The first few Fibonacci numbers are (starting with 0 and 1): 0, 1, 1, 2, 3, 5, 8 ,13 , 21 ….

The Fibonacci Series is the sum of values of the previous n values in the Fibonacci sequence. For example, 1+2=3; so start at n=0 and do the same thing to get the next value. 0+1=1; 1+1=2; 2+3=5; so you would write this series as: 0+1+1+2+3 = 5 then do that for all numbers starting with n =0 . The first few terms of this sequence are: 0, 1, 2, 3, 5, 8 ,13 ,21 .

The Fibonacci sequence is used in several areas of mathematics including geometry and biology. It is also used in many daily activities like pricing problems and stock trading. Although it can be seen naturally in many things like the spiral on a pinecone or the arrangement of leaves on a stem.

The Fibonacci numbers are useful because they form one of two different number series that provide accurate approximations of π (3.1415…).

The fibonacci series can be represented by the Binomial Coefficient. It is also known as “choose” and written with an X in math notation. It represents the number of ways you can choose some things from a group without order and repetition or replacement. For example: choosing 2 out of 4 objects (4 total) means there are 24 different combinations possible.

So, you would write the Fibonacci Series like this: (n)(X^(n-2)) *(-1) ^(n-1)

So, if n =5 then it looks like this: (5) (X^ (5-2)) *(-1) ^4

## Fibonacci Numbers and Their Properties

Many computer methods, such as Fibonacci cubes, Fibonacci heap storage structures, and the fibonacci search methodology, use Fibonacci numbers. Let’s look at the Fibonacci numbers’ varied features dependent on their position above and below zero.

– Fib(n) = 0 if n=0 or -1

– Fib(n) = 1 if n=1

– Fib(n) = Fib(n-1)+Fib(n-2) if n>0; not including the endpoints. (thus, for example, Fib(-3)=Fib(-2)+Fib(-4).)

## Fibonacci Numbers and Golden Ratio Relationship

When two fibonacci numbers are added together, the ratio is quite near to 1.618034. Let’s look at an example of two consecutive numbers at random:

Let’s take A = 13, B = 21, and divide B by A. 21 13 = 1.625 is the result.

The Golden Ratio is the ratio of successive Fibonacci numbers.

## Nature’s Fibonacci Numbers

·       Fibonacci numbers can be seen all across nature. Here are some of the most prevalent Fibonacci number patterns and sequences found in nature:

• Fibonacci flowers are blooms that follow the Fibonacci pattern and are found in plants such as sunflowers, lilies, roses, and buttercups.
• Fibonacci numbers are represented by the spirals on the pinecone.
• Sunflower seeds are reported to follow a Fibonacci pattern as well.
• Seashells and starfish that we find on the beach match the Fibonacci number pattern.

## A Real-life Example of the Fibonacci Series

According to the Fibonacci numbers in flowers the center disk of yellow or orange contains florets arranged in two Fibonacci arcs (arcs 1 and 2). The number of these arcs determines whether it is an odd-numbered or even-numbered flower head. Those with nine and sixteen florets follow another example of consecutive Fibonacci numbers, while those with thirty-four and fifty-five florets follow yet another example.

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