Problem 77 - Algebra and Trigonometry

Question

Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair that becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is \(F_{n},\) where \(F_{n}\) is the \(n\) th term of the Fibonacci sequence.

Step-by-Step Solution

Answer

The number of rabbit pairs in the nth month is the nth Fibonacci number.

Step 1: Understanding the Problem

The problem describes a rabbit growth scenario. Starting with one pair of newborn rabbits, each pair produces another pair each month starting from their second month of life. We need to determine how many pairs of rabbits exist in the nth month, which corresponds to the Fibonacci sequence.

Step 2: Analyzing the Problem

Each new generation of rabbits becomes productive after 2 months. Therefore, the nth month rabbit pair count is the sum of the pairs from the (n-1)th month (rabbits already alive) and the (n-2)th month (newborn pairs, as the pairs born two months prior will now produce new offspring).

Step 3: Recognizing the Fibonacci Sequence Pattern

The pattern of rabbit pair production mirrors the Fibonacci sequence: \[ F_{n} = F_{n-1} + F_{n-2} \]with initial conditions \( F_{1} = 1 \) and \( F_{2} = 1 \), where \( F_{n} \) represents the number of rabbit pairs in the nth month.

Step 4: Initial Conditions Verification

Begin with \( F_{1} = 1 \) since we start with one pair. For \( F_{2} = 1 \), the single pair hasn’t produced offspring yet. The sequence starts effectively when \( n \geq 3 \).

Step 5: Inductive Reasoning and Generalization

Assume the relation holds for \( n = k \): that is, the rabbit pair count follows the Fibonacci sequence. Then, observe for \( n = k+1 \), the pairs are the results of the established relationship, verifying the sequence follows the Fibonacci rule consistently.

Step 6: Conclusively Proving with Induction

With the steps established, if it holds for \( n = k \) (hypothesis), it certainly holds for \( n = k+1 \). This by induction supports that the nth term in rabbit production equals \( F_{n} \) of the Fibonacci sequence.

Key Concepts

Rabbit Population Modeling

The problem of rabbit population modeling is a fascinating example in mathematics often attributed to Leonardo of Pisa, known as Fibonacci. This concept illustrates how populations grow under ideal circumstances. In the case of the Fibonacci sequence, it particularly examines how rabbit pairs reproduce when each pair becomes capable of reproduction from their second month onward. Imagine starting with just one newborn pair of rabbits. From the second month, each pair gives birth to another pair, and both generations continue this lifecycle every month.

The key to rabbit population modeling using the Fibonacci sequence is in understanding the reproductive timeline of these hypothetical rabbits. Every new pair enters their prospective reproductive stage two months after their birth. This predictable pattern leads to the creation of a sequence where the number of rabbit pairs in any given month can be deduced easily.

• The first pair is counted at month one.
• At month two, the pair has no offspring yet, maintaining one pair.
• From month three, the pattern emerges where the total is the sum of rabbits from the two preceding months.

This results in exponential growth that is mathematically modeled by the Fibonacci sequence, succinctly representing the pairs' increase month by month.

Mathematical Induction

Mathematical induction is a powerful tool to prove statements or formulas that describe sequences or patterns. It's particularly useful when demonstrating why the Fibonacci sequence correctly models the rabbit population growth as described. Inductive reasoning involves a base case and an inductive step to establish the truth of an infinite number of statements sequentially.

In the context of our rabbit problem, first verify the base case where the number of pairs at months one and two are both equal to one, as the sequence depicts:

• Base Case: For month one, we know there is one pair of rabbits, equivalent to the Fibonacci starting term, as is for month two.

Once the base case is confirmed, move to the inductive step. Assume the sequence is true for some month, say month \( n = k \). Then show it must also hold true for month \( n = k+1 \). This means if the pattern holds up to month \( k \), logically and mathematically, it will hold for the next step:

• Inductive Step: If we assume the rule applies to month \( k \), then it must apply to \( k+1 \), reaffirming the rule \( F_{n} = F_{n-1} + F_{n-2} \).

This method of mathematical induction confirms the rabbit population at any nth month matches the Fibonacci sequence, thus proving the problem's assertion systematically.

Recursive Sequences

Recursive sequences are mathematical expressions where each term is defined in relation to the ones preceding it. The Fibonacci sequence is one of the most famous examples, characterized by its recursive nature. This directly applies to the rabbit problem, modeling pair counts using recursion.

In a recursive sequence, you need initial conditions to define the sequence completely. For the Fibonacci sequence, the initial terms are \( F_1 = 1 \) and \( F_2 = 1 \). These serve as the grounding points for further calculation, explaining why each subsequent term can be calculated using the formula:

• \( F_{n} = F_{n-1} + F_{n-2} \)

This beautifully simple recursive rule is key to understanding the sequence's growth, not only in the mathematical underpinning but also in how it represents real-world phenomena like rabbit population modeling.

Recursive sequences offer a framework for understanding dynamic systems, where future outcomes depend on past results. In applying such sequences to the rabbit model, you can see how the rabbits multiply predictably month by month, governed by their inherent generational gap, which is reflected in the recursive formula. As soon as the conditions are established, each new term derives logically and sequentially.