Problem 77 - Precalculus: Mathematical for Calculus

Question

Fibonacci's Rabbits 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 nth month is the nth Fibonacci number, \( F_n \).

Step 1: Understanding the Fibonacci Sequence

The Fibonacci sequence is a series of numbers in which each number (after the first two) is the sum of the two preceding numbers. The sequence typically starts with 0 and 1. So the sequence is: 0, 1, 1, 2, 3, 5, 8, 13, ... where each subsequent number is the sum of the previous two.

Step 2: Define the Initial Conditions

For rabbit pairs, start with one pair at month 1: \( F_1 = 1 \). In month 2, we still have one mature pair, which hasn't reproduced yet: \( F_2 = 1 \).

Step 3: Develop the Recursive Formula

From month 3 onwards, each pair that existed two months ago can now reproduce. Hence, the number of rabbit pairs at month \( n \) is equal to the number of pairs in month \( n-1 \) plus the number of pairs in month \( n-2 \). This gives the recursive formula: \( F_n = F_{n-1} + F_{n-2} \).

Step 4: Verify the Formula

To verify the formula, compute the first few terms explicitly:- Month 1: 1 pair \( (F_1 = 1) \)- Month 2: 1 pair \( (F_2 = 1) \)- Month 3: 2 pairs \( (F_3 = 2 = F_2 + F_1) \)- Month 4: 3 pairs \( (F_4 = 3 = F_3 + F_2) \)- Month 5: 5 pairs \( (F_5 = 5 = F_4 + F_3) \)- and so on.Thus, it follows the Fibonacci sequence.

Step 5: Conclude with the General Solution

In conclusion, the number of rabbit pairs at month \( n \) is exactly the \( n \)th term of the Fibonacci sequence \( F_n \). This is because the conditions and growth model precisely mimic the definition and evolution of the Fibonacci sequence.

Key Concepts

Recursive Formula

The concept of a **recursive formula** is pivotal in many mathematical sequences, including the famous Fibonacci sequence. A recursive formula is like a set of instructions or a recipe. It helps you find any term in a sequence if you know the preceding ones. In the Fibonacci sequence, each term after the first two is the sum of the two preceding terms. This is expressed mathematically as:

\[F_n = F_{n-1} + F_{n-2},\ \text{for} \ n \geq 3\]

• **Recursive nature**: Every term depends on the two terms before it. This gives it a chain-like reaction effect amusingly common in nature.
• **Computation**: You start with known values, then apply the recursive formula to find later terms.

Understanding this formula is crucial for calculating any term in the Fibonacci sequence. It mirrors natural processes like rabbit reproduction, leading to the classic rabbit problem posed by Fibonacci himself.

Initial Conditions

In solving problems that involve sequences, defining **initial conditions** is crucial. These are the starting points which allow us to apply the recursive formula effectively.
Let's examine the initial conditions of the Fibonacci sequence as posed in the rabbit problem:

• **Month 1**: Start with one pair of newborn rabbits. Hence, \( F_1 = 1 \).
• **Month 2**: The rabbits are mature but have not reproduced yet. So, \( F_2 = 1 \).

These initial conditions are the building blocks for the Fibonacci sequence. By specifying the first two months, it directly sets the stage for using the recursive formula to predict rabbit numbers in subsequent months.

Sequence Terms

The **sequence terms** are really the framed 'results' of the Fibonacci problem. Each term in a sequence has a distinct position and value, defined by the recursive relation and initial conditions.
To illustrate, let's see the first few terms in the Fibonacci sequence:

• **Term 1**: One pair of rabbits (\( F_1 = 1 \)).
• **Term 2**: Still one pair (\( F_2 = 1 \)).
• **Term 3**: Two pairs, as they start reproducing (\( F_3 = 2 \)).
• **Term 4**: The herd grows to three pairs (\( F_4 = 3 \)).
• **Term 5**: Expand to five pairs (\( F_5 = 5 \)).

Each term is interconnected. The value of each term not only has meaning in the context of rabbit population growth but also fits into the pattern dictated by the Fibonacci sequence. These terms highlight how mathematical sequences can model real-world phenomena accurately.