Leonardo Pisano, called Fibonacci, proposed an interesting question in his book Liber Abaci: A pair of newly born rabbits is brought into a confined place. This pair, and every later pair, begets one new pair every other month, starting in their second month of age. How many pairs will there be after one, two, ... months, assuming that no deaths occur?
For the first two months, we would have no new rabbits, but at the start of the third month, we would have a new pair, and a new pair in the fourth month. Generally, if there are F(n) pairs a the start of month n, and F(n+1) pairs at the start of month n+1. The latter will still be there at the beginning of the (n+2)th month, but so will the new pair of offspring of every pair that existed in month n. Thus,
F(n+2) = F(n+1)+F(n)
This sequence yields this:
| n | F(n) |
| 1 | 1 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 5 |
| 6 | 8 |
| 7 | 13 |
This is known as the Fibonacci sequence, and as we will see, it is found almost everywhere. Next we will look at the relationship between the Fibonacci sequence and the Golden ratio.