Chapter 2.2, Problem 6MS

Baby bunnies. This question gave the Fibonacci sequence its name. It was posed and answered by Leonardo of Pisa, better known as Fibonacci.

Suppose we have a pair of baby rabbits: one male and one female. Let us assume that rabbits cannot reproduce until they are one month old and that they have a one-month gestation period. Once they start reproducing, they produce a pair of bunnies each month (one of each sex). Assuming that no pair ever dies, how many pairs of rabbits will exist in a particular month?

During the first month, the bunnies grow into rabbits. After two months, they are the proud parents of a pair of bunnies. There will now be two pairs of rabbits: the original, mature pair and a new pair of bunnies. The next month, the original pair produces another pair of bunnies, but the new pair of bunnies is unable to reproduce until the following month. Thus we have:

Continue to fill in this chart and search for a pattern. Here is a suggestion: Draw a family tree to keep track of the offspring.

We’ll use the symbol F₁ to stand for the first Fibonacci number, F₂ for the second Fibonacci number, F₃, for the third Fibonacci number, and so forth. So ${\text{F}}_{\text{1}}\text{=1,}$ and ${\text{F}}_2\text{=1,}$ and, therefore, ${\text{F}}_3{\text{=F}}_{\text{2}}{\text{+F}}_{\text{1}}\text{=2,}$ and ${\text{F}}_4{\text{=F}}_3{\text{+F}}_2\text{=3,}$ and so on. In other words, we write F_n for the nth Fibonacci number where n represents any natural number; for example, we denote the 10th Fibonacci number as F₁₀ and hence we have ${\text{F}}_{\text{10}}\text{=55}\text{.}$

So, the rule for generating the next Fibonacci number by adding up the previous two can now be stated symbolically, in general, as: