Fibonacci
Bewijs:
The number of rabbit couples after n months will be denoted as fn.
We will show that fn = fn-1 + fn-2.
Denote the number of newborn rabbit couples after n months by pn and
the number of not-newborn couples after n months by gn.
Then fn = gn + pn.
A month later there will be born pn+1 couples. Each couple
is produced by a couple that was not newborn one year ago.
There are gn of them.
Apparently is pn+1 = gn.
The fn couples after n months are the not-newborn couples after n+1 months.
So, fn = gn+1
We have obtained the following relations:
1. fn = gn + pn
2. pn+1 = gn and
3. fn = gn+1
Thus: fn = gn + pn =
(from 2.) gn + gn-1 = (from 3.)
fn-1 + fn-2.
And that is what we wanted to show.