Problem 85 - Precalculus: Mathematics for Calculus

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 \(n\)th 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 \(n\)th month is \(F_{n}\), the nth term of the Fibonacci sequence.

Step 1: Understand the Problem

The problem given is a classic Fibonacci sequence problem. It asks us to determine the number of rabbit pairs after \(n\) months, considering the reproductive rules. Every pair becomes reproductive after 2 months and thereafter produces a new pair each month.

Step 2: Define the Fibonacci Sequence

The Fibonacci sequence is defined as \(F_0 = 0\), \(F_1 = 1\), and \(F_{n} = F_{n-1} + F_{n-2}\) for \(n \geq 2\). This sequence represents how each term is derived by adding the two preceding terms.

Step 3: Relate the Problem to Fibonacci Sequence

We start with one pair of newborn rabbits (\(F_1 = 1\)). In the next month, they grow but cannot reproduce (\(F_2 = 1\)). The reproduction rule implies that any new pair becomes productive only two months after birth. From the second month onwards, each productive pair produces a new pair, following the equation \(F_{n} = F_{n-1} + F_{n-2}\).

Step 4: Establish Initial Conditions

We know the rabbits start with one pair at month 1, i.e., \(F_1 = 1\), and remain with one pair at month 2, i.e., \(F_2 = 1\). Use these as base cases for induction in the Fibonacci sequence.

Step 5: Use Induction to Prove General Formula

Assuming that \(F_{k}\) represents the number of rabbit pairs in month \(k\), by induction, we assume \(F_{n-1}\) and \(F_{n-2}\) are true. Then, by the breeding rule, we have \(F_{n} = F_{n-1} + F_{n-2}\), showing that \(n\)th month rabbits follow the Fibonacci sequence.

Step 6: Conclude that \(F_{n}\) Represents Rabbit Pairs

From induction, each month corresponds exactly to a term in the Fibonacci sequence: the number of rabbit pairs in month \(n\) is exactly the \(n\)th term \(F_n\) of the Fibonacci sequence.

Key Concepts

Inductive Reasoning

Inductive reasoning is a method of reasoning in which we go from specific instances to a general conclusion. It's often used in mathematics to show that a statement holds true for all natural numbers. In our rabbit problem, inductive reasoning helps us prove the general formula for the number of rabbit pairs each month.

To apply this to the Fibonacci rabbit problem, we start by establishing a base case. Here, the starting scenario is our base case: At month 1 and month 2, there is exactly one pair of rabbits, which corresponds to the Fibonacci numbers \(F_1=1\) and \(F_2=1\).

• Base Case: To prove by induction, we show it holds for \(n=1\), \(n=2\).
• Inductive Step: Assuming it holds for \(n=k\) and \(n=k+1\), we show it holds for \(n=k+2\).

During the inductive step, we assume that the number of rabbit pairs at months \(k\) and \(k+1\) follows the Fibonacci pattern. This assumption lets us conclude the number of rabbit pairs at month \(k+2\) will be the sum of the previous two terms, which aligns perfectly with how the Fibonacci sequence is constructed.

This method assures us that our hypothesis is true universally for any time \(n\), thanks to the logical foundation provided by induction.

Mathematical Modeling

Mathematical modeling involves representing real-world situations with mathematical constructs. It aids in understanding complex phenomena by simplifying them into manageable components. With our rabbit problem, mathematical modeling illustrates how the Fibonacci numbers model the rabbit population over time.

The rabbit reproduction scenario can be elegantly described using the rules of the Fibonacci sequence. Here’s how these elements translate:

• Rabbits produce offspring, analogous to accumulating values in the Fibonacci series.
• Each pair becomes productive after a set period, reflected by the sequence's recurrence relations.

This approach translates the reality of rabbit reproduction into a sequence of numbers, making complex biological processes easier to analyze and predict.

Mathematical modeling helps us abstract and solve the problem more effectively, reducing the world’s complexity to neat numerical sequences. It shows how the Fibonacci sequence not only matches this animal scenario but also appears in other natural phenomena, from nautilus shells to sunflower seed arrangements. Such modeling provides deep insights into the rules governing natural patterns.

Recurrence Relations

Recurrence relations are fundamental tools in discrete mathematics. They define sequences recursively, where each term is a function of its predecessors. The Fibonacci sequence is one of the classic examples of a recurrence relation.

The given problem perfectly encapsulates a recurrence relation:

\[F_n = F_{n-1} + F_{n-2}\]

where each number after the first two approximates the sum of the two preceding numbers. It's a potent way to characterize the sequence's growth without needing an explicit closed formula.

• Base Cases: Establish initial conditions, here being \(F_0 = 0\) and \(F_1 = 1\).
• Recursive Formula: Defines how each term of the sequence relates to those before it.

Applying this to our rabbit scenario, the number of rabbit pairs each month arises from the sum of the pairs that existed in the two previous months. Thus, recurrence relations help translate a simple breeding principle into a structured numerical sequence.

This mathematical framework not only captures the essence of growth in populations but also finds applications in algorithm design, financial analyses, and various natural sciences.