Understanding the Fibonacci Sequence

 Fibonacci Sequence 


It is said that the golden ratio itself springs from the Fibonacci sequence, so it is important to first understand the importance and the meaning of the Fibonacci Sequence. 

What is the Fibonacci Sequence? 

In simplest words, it is a naturally occurring sequence or series of numbers, in which each new number is added to the one that precedes it. It can also be described as the sum of two numbers before it. For example, if it begins with 0, and 1 is added, we get 1, then when 1 is added again, we get two. And the sum of 1 and 2 is 3, the sum of 2 and 3 is 5, and the sum of 3 and 5 is 8. 
So we can say that it goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and so on. It doesn't necessarily have a limit and may go on till infinity. 

There are many debates regarding the origin of the Fibonacci sequence, as some sources associate it with the Greeks, some claim it was an Indian discovery, and some argue that it is an Italian discovery, as the Fibonacci numbers were named after an Italian mathematician from Pisa, named Leonardo. This is because it was he who unveiled the concept of the sequence to Western European mathematics. 




Picture Source: Google Images. 


The Fibonacci sequence in nature can be seen in leaves in the trees, the bracts of pinecone, shapes of shells, seed heads, and branches of trees. 






The Fibonacci sequence frequently appears in mathematics, and computer programs and algorithms too rely on it. 

HISTORY:

Fibonacci, in his book, "Liber Abaci", introduced the concept while calculating rabbit population's growth. (Given the rabbits would mate at age one month and gestate for a month) He initiated with a single pair of rabbits put on a field, who eventually mate by the end of the first month, and produce a pair by the end of the second month. Then the original/initial pair mate again and produce a second pair of rabbits, and there are 3 pairs now (the first produced pair has not yet reproduced). By the end of the third month, the original pair produce a new (third) pair, and the first born pair produce a pair, so the total pairs on the farm now are 5. By the end of the 4th month, the original pair and the first two born pairs would have reproduced, resulting in 8 pairs of rabbits (5+3).  

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