How did Fibonacci find the Fibonacci sequence?

Answer 1

It is said that he was modelling rabbit population. Each pair of rabbits needed to be two periods old before they could breed and then they would produce a pair of rabbits. The first lot of rabbits then would keep producing a fresh pair of rabbits every period after that - forever! The newly born rabbits would be juveniles until they reached period 2 when they, too, would join the breeding population.

Start with 1 pair in period 1. In period 2 they are still juveniles. In period 3 they breed 1 pair. So that makes two pairs. In period 4, they breed again but their offspring are still juveniles so one more pair which makes 3 pairs. In the next period, the original pair breed , and their first set of offspring breed so 2 more pairs, and so on.

This gives rise to the recursive formula for the Fibonacci sequence:

t(1) = 1

t(2) = 1

and t(n) = t(n-2) + t(n-1) for n > 2