Statement of the Problem:

A problem posed by Leonardo Fibonacci in the early 13th century asks how many rabbits you would have if you started with just two and let them reproduce. The conditions are that a pair of rabbits produces a pair of offspring each month, and each pair produces their first offspring at the age of 2 months.

Stage 1: Understanding the Problem:

We need to determine the number of pairs of rabbits month by month. The key observation is that the number of pairs in any month is given by the sum of rabbits in the two preceding months, except for the 0th and 1st months where we start with 1 pair each.

Stage 2: Mathematical Description:

We formulate the problem mathematically. If we denote the number of pairs in the nth month as $F(n)$, then:

• $F(0) = 1$
• $F(1) = 1$
• $F(n) = F(n-1) + F(n-2)$ for $n \geq 2$

This is the famous Fibonacci sequence.

Stage 3: Solving the Mathematical Problem:

Using the recurrence relation, we compute the values step by step:

$$F(2) = 1 + 1 = 2$$ $$F(3) = 2 + 1 = 3$$ $$F(4) = 3 + 2 = 5$$

and so on. Continuing this process up to $F(16)$, we get 1597 pairs.

Stage 4: Interpretation:

The mathematical solution shows that after just 16 months, we have nearly 1600 pairs of rabbits. This demonstrates how quickly the population grows under the given assumptions.