Chapter 10 The Fibonacci Sequence
In Liber Abaci (around 1202), Fibonacci asked the following question.
A man put one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year, if the nature of these rabbits is such that every month each pair bears a new pair which from the second month on becomes productive?
In the first month we have \(1\) adult pair bearing \(1\) new pair for a total of \(2\) pairs. In the second month we have \(2\) adult pairs, only one of which can bear a new pair, for a total of \(3\) pairs. In the third month we have \(3\) adult pairs only \(2\) of which can bear a new pair, for a total of \(5\) pairs. Continuing in this manner we concern ourselves with the sequence
We can define this sequence recursively, denoting \(f_n\) the \(n\)th Fibonacci number, by \(f_1 = 1, f_2 = 1,\) and for \(n\geq 3\) we have
The Fibonacci sequence is our first instance of a recursive sequence, and in this sense it is interesting apart from its connection to the breading habits of hypothetical rabbits. For this reason we make a more abstract investigation of the sequence \(\{f_n\}_n.\)