This problem is from Fibonnaci’s Liber Abaci, written in the year $1202$. Imagine you have a pair of young rabbits (a male and a female) and after a month they mature into adult rabbits. Every month after that they breed and produce another young pair of rabbits. The young rabbit pairs similarly take a month to mature, then start producing another pair of young rabbits every month. So in the first month we have $1$ pair, in the second month still only $1$ pair (now adults), in the third month $2$ pairs ($1$ adult, $1$ young), in the fourth month $3$ pairs ($2$ adult, $1$ young), in the fifth month $5$ pairs ($3$ adult, $2$ young), and so on. Find the number of pairs of rabbits in the twelfth month. Show answer
In the sixth month, the $3$ adult pairs produce young, so we have $5 + 3 = 8$ pairs ($5$ adult, $3$ young). In the sixth month, the $5$ adult pairs produce young, so we have $8 + 5 = 13$ pairs ($8$ adult, $5$ young). In each month, the number of new pairs is the same as the number of adult pairs in the last month, which is the total number of pairs in the month before that. So each month, the number of pairs is the number in the previous month plus the number in the month before that. Now that we know the pattern, it’s easy to fill in the rest: $1,1,2,3,5,8,13,21,34,55,89,144,\ldots$, so there are $144$ pairs of rabbits in the twelfth month.
This list of numbers is called the Fibonacci sequence, although Fibonacci was not the first person to explore it. Question 13.2.3 describes the conundrum that lead to the first record of the sequence.