On The Divisibility Property Of Fibonacci Sequence

The method of generating functions is one of the most useful tools of the trade of enumerative combinatorics.Generating functions are so useful because information about a sequence translates to information about function that is often easier to process,getting additional information about function that often translates back to information about the sequences.There are many important applications,such as deducing recursive relation, seeking the mean value of the sequence and proving unimodality.Fibonacci sequence and it's properties are one of the most interesting problems in combinatorics.In virtue of this method,we prove the divisibility property of Fibonacci sequence in this paper.The main content of this thesis can be summarized as follows:1.In chapter 1,we introduce the origin and development of Fibonacci number and recapitulate the work of this article.2.In chapter 2,we introduce the main method used in this paper:the generating functions. By means of the viewpoint of abstract algebra,the generating functions are defined as formal power series.All the formal power series can form an integral domain after introducing a kind of addition and multiplication of the formal power series.It gives a rigorous theoretical basis to the four arithmetic operation of the generating function. Then we introduce two kinds of formal power series generating functions in detail.In the end,we display vividly some concrete applications the method of generating functions.3.In chapter 3,we give the ordinary power series generating function of the sequence {F_mn}_n=0^∞,for any positive integer m.Furthermore,we prove the divisibility property of Fibonacci sequence.