Consider the following situation. A problem dating back to the early 13th century, posed by Leonardo Fibonacci asks how many rabbits you would have if you started with just two and let them reproduce. Assume that a pair of rabbits produces a pair of offspring each month and that each pair of rabbits produces their first offspring at the age of 2 months. Month by month the number of pairs of rabbits is given by the sum of the rabbits in the two preceding months, except for the 0th and the 1st months. After just 16 months, you have nearly 1600 pairs of rabbits! Clearly state the problem and the different stages of mathematical modelling in this situation.
Statement of the Problem: A problem posed by Leonardo Fibonacci in the early 13th century asks how many pairs of rabbits you would have if you started with just two and let them reproduce. The conditions are: 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 find the number of pairs of rabbits month by month under the given reproductive conditions. We start with 1 pair at month 0 and month 1.
Stage 2 - Mathematical Description: The problem translates to finding a pattern where the number of pairs in any month equals the sum of rabbits in the two preceding months. This gives us the sequence:
where each term (after the first two) is the sum of the two previous terms.
Stage 3 - Solving the Mathematical Problem: Using the rule that each term equals the sum of the two preceding terms, we can calculate the number of pairs for each month. The table shows values from month 0 (1 pair) to month 16 (1597 pairs).
Stage 4 - Interpretation: After 16 months, we have nearly 1600 pairs of rabbits. The model shows how a simple reproductive rule leads to rapid population growth, demonstrating the famous Fibonacci sequence.
Explanation
This question from Exercise A2.1 tests students' understanding of mathematical modelling stages. The context provides the complete Fibonacci sequence table and references the modelling process shown in Fig. A2.1. Students should identify that mathematical modelling involves: understanding the problem, creating a mathematical description, solving it, and interpreting results. The Fibonacci sequence emerges naturally from the rabbit reproduction rules.
Solution Steps
Step 1: State the problem clearly with all given conditions
Step 2: Identify Stage 1 - Understanding what we need to find
Step 3: Identify Stage 2 - Describe the mathematical relationship (sum of two preceding months)
Step 4: Identify Stage 3 - Show how the sequence is calculated
Step 5: Identify Stage 4 - Interpret the result (nearly 1600 pairs after 16 months)