Problem of rabbits reproduction Leonardo Pisano Fibonacci is considered rightfully as one of the most known mathematicians of the Middle Ages epoch. Later we will tell about Fibonacci and his role in development of the West-European mathematics in more detail. By irony of his fate Fibonacci introduced the outstanding contribution to development of mathematics, there was known in modern mathematics only as the author of interesting numeric sequence called Fibonacci numbers. This numeric sequence was obtained by Fibonacci at the solution of the famous The essence of his "problem of rabbits reproduction" Fibonacci formulated extreme simply: "Let's suppose that in the fenced place there is the couple of the rabbits (female male) in first day of January. This couple of the rabbits reproduces the new rabbits couple in the first day of February and then in the first day of each next month. Each newborn rabbits couple becomes mature in one month and then gives a life to the new rabbits couple each month after. There is a question: how much rabbits couples will be in the fenced place in one year, that is in 12 months from the beginning of reproduction?" For the solution of this problem, which is demonstrated in figure above we will designate through
Let's note, that the passage of (1) models the monthly transformation of each mature rabbits couple in two couples, namely in the same pair of the mature rabbits
Note that in the columns Studying the
Such formula is called as the Note that the concrete values of the numeric sequence generated by the recurrent formula (3), depend on the initial values of the sequence F. For example, we have _{2}F = _{1}F = 1 for _{2}A-numbers and for this case the recurrent formula (3) "generates" the following numeric sequence:
For the F = 1; then the corresponding numeric sequence for this case will look as the following:_{2}0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... . At last, for the ( F = 2; then the conforming numeric sequence for this case will look as the following:_{2}1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... . In mathematics the numeric sequence of (4) is called, as a rule, Fibonacci numbers. They have a number of surprising mathematical properties, and we will tell about them at the next pages of our Museum. Follow us! |