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Fibonacci Sequence and Rabbit Problem

Hello friends hope you all are good , healthy and safe. 

In this Blog we will learn about fibonacci Sequence , golden ratio,relation between them , rabbit problem and many more.


Also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
Fibonacci 

Greatest mathematician of the middle ages, Fibonacci. Fibonacci was born in pisa around 850 years ago. He was a famous Italian mathematician. In the year 1202 he finished his book "Liber Abeci".

The book of calculations that brought Arabic numerals to Europe, the zero one,two that we use today. Fibonacci also presented a growing rabbit population problem. And after solving this problem he derived the famous sequence of numbers that is named after him, The Fibonacci Sequence .

Recursion relation to find all the Fibonacci numbers using initial two numbers. Also observed the Fibonacci spiral which you will find in nature also.

Suppose you put a male-female pair of newly born rabbits in a field . Rabbits take a month to mature before mating. After one month , females gives birth to one male-female pair and then mate again. No rabbits die. How many rabbit pairs are there after one year?


To solve this, we construct table. At the start of each month, the number of young pairs, adult pairs, and total number of pairs are shown. At the start of January, one pair of young rabbits is introduced into the population. At the start of February, this pair of rabbits has became adult. At the start of March, this pair has given birth to a new pair of young rabbits. And so on.


  Observation Table
                          
The observation table shows Fibonacci’s rabbit population in each month of the year.
Now we define the Fibonacci numbers Fn to be the sum of total number of rabbit pairs at the start of the nth month. The number of rabbits pairs at the start of next year or the 13th month i.e january, F13 = 233, can be taken as the solution to Fibonacci’s puzzle.

Further observing the table of the Fibonacci numbers listed in observations Table , revels or discovers that these numbers satisfy the recursion relation given below,

Fn+1 = Fn + Fn−1

This recursion relation gives the next Fibonacci number as the sum of the preceeding two numbers.
To start this recursion sequence , we need to specify F1 and F2 that is initial data . In this Fibonacci’s rabbit problem, the initial month starts with only one rabbit pair so that F1 = 1.  This initial rabbit pair is newborn and young so they takes one month to mature before mating therefore F2 = 1.
The first few Fibonacci numbers, read from the table, are given by
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
and has become one of the most famous sequences in mathematics.




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