The Medieval Mathematician Leonardo Fibonacci's Problem About Rabbits

Let's see how the number of rabbits grows in the first six months:

Month 1. One pair of young rabbits.

Month 2. There is still one original pair. Rabbits have not yet reached childbearing age.

Month 3. Two pairs: the original, reaching childbearing age + a pair of young rabbits that she gave birth to.

Month 4. Three pairs: one original pair + one pair of rabbits that she gave birth to at the beginning of the month + one pair of rabbits that were born in the third month, but have not yet reached sexual maturity.

Month 5. Five pairs: one original pair + one pair born in the third month and reached childbearing age + two new couples that they gave birth to + one couple, which was born in the fourth month, but has not yet reached maturity.

Month 6. Eight couples: five couples from last month + three newborn couples. Etc.

To make it clearer, let's write the received data into the table:

If you carefully examine the table, you can identify the following pattern. Each time the number of rabbits present in the nth month is equal to the number of rabbits in the (n - 1) th previous month, summed up with the number of newly born rabbits. Their number, in turn, is equal to the total number of animals as of the (n - 2) month (which was two months ago). From here you can deduce

formula:

F_n = F_{n ‑ 1}+ F_{n ‑ 2},

where F_n - the total number of pairs of rabbits in the n-th month, F_{n ‑ 1} Is the total number of pairs of rabbits in the previous month, and F_{n ‑ 2} - the total number of pairs of rabbits two months ago.

Let's count the number of animals in the following months using it:

Month 7. 8 + 5 = 13.

Month 8. 13 + 8 = 21.

Month 9. 21 + 13 = 34.

Month 10. 34 +21 = 55.

Month 11. 55 + 34 = 89.

Month 12. 89 + 55 = 144.

Month 13 (beginning of next year). 144 + 89 = 233.

At the beginning of the 13th month, that is, at the end of the year, we will have 233 pairs of rabbits. Of these, 144 couples will be adults and 89 will be young. The resulting sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 called Fibonacci numbers. In it, each new final number is equal to sum the previous two.