Problem 83 - Calculus: Early Transcendentals

Question

(a) Fibonacci posed the following : Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs or rabbits will we have in the \( n \)th month? Show that the answer is \( f_n \) where \( \\{ f_n \\} \) is the Fibonacci sequence defined in Example 3(c). (b) Let \( a_n = f_{n + 1} / f_n \) and show that \( a_{n - 1} = 1 + 1/a_{n - 2}. \) Assuming that \( \\{ a_n \\} \) is convergent, find its limit.

Step-by-Step Solution

Answer

(a) In month \( n \), there are \( f_n \) rabbit pairs. (b) The limit of \( \{a_n\} \) is the golden ratio, \( \frac{1 + \sqrt{5}}{2} \).

Step 1: Understanding the Fibonacci Sequence Conditions

According to the problem, the number of rabbit pairs each month corresponds to the Fibonacci sequence. The rule states that each pair of rabbits reproduces a new pair each month starting from the 2nd month. The first few terms of the Fibonacci sequence are: 1 (month 1), 1 (month 2), 2 (month 3), 3 (month 4), 5 (month 5), and so on.

Step 2: Formulating the Fibonacci Sequence

Define the Fibonacci sequence as follows: \( f_1 = 1 \), \( f_2 = 1 \), and for \( n \geq 3 \), \( f_n = f_{n-1} + f_{n-2} \). This sequence fits the problem condition exactly, as it reflects how the number of rabbit pairs grow each month.

Step 3: Relating Fibonacci to Rabbits

In the \( n \)th month, the number of pairs of rabbits, starting with one newborn pair, will be \( f_n \) due to the recursive nature of their reproduction pattern aligning with the Fibonacci sequence.

Step 4: Defining the Sequence for Ratios

We define another sequence \( a_n = \frac{f_{n+1}}{f_n} \). The task is to express \( a_{n-1} \) in terms of previous terms and show that \( a_{n-1} = 1 + \frac{1}{a_{n-2}} \).

Step 5: Using Fibonacci Properties

Based on the property of Fibonacci numbers, express \( f_{n+1} = f_n + f_{n-1} \) and similarly for other terms in the ratio expression.

Step 6: Manipulating the Sequence Expression

Rewrite \( a_{n-1} = \frac{f_n}{f_{n-1}} \). Express\( \frac{f_{n-1}+f_{n-2}}{f_{n-2}} \) using Fibonacci recursive relationships to find \( a_{n-1} = 1 + \frac{f_{n-2}}{f_{n-1}} = 1 + \frac{1}{a_{n-2}} \).

Step 7: Finding the Limit of the Sequence

Assuming the sequence \( \{ a_n \} \) converges to \( L \), equate \( L = 1 + \frac{1}{L} \). This simplifies to the quadratic equation \( L^2 = L + 1 \), resulting in \( L \) being the positive solution, \( L = \frac{1 + \sqrt{5}}{2} \), the golden ratio.

Key Concepts

Recursive Relationships

Recursive relationships are a fascinating concept in mathematics, where each term in a sequence is defined based on the previous terms. This idea is beautifully illustrated in the Fibonacci sequence. Each term is the sum of the two preceding ones, creating a chain of relationships that continue indefinitely.
For the Fibonacci sequence, the rule can be written as:

• \( f_1 = 1 \)
• \( f_2 = 1 \)
• For \( n \geq 3 \), \( f_n = f_{n-1} + f_{n-2} \)

These relationships explain how the number of rabbit pairs grows over months according to Fibonacci's original rabbit problem. Starting with one pair, the count of rabbit pairs in any month is the sum of the pairs from the two preceding months, making the sequence grow exponentially in a structured manner.

Mathematical Induction

Mathematical induction is a powerful proof technique used to prove statements about natural numbers. It's like a domino effect – if we can show that a property holds for an initial case and that if it holds for one case, it holds for the next, we can conclude it holds for all cases.
Consider the sequence of rabbit pairs given by the Fibonacci sequence. To formally establish the sequence's accuracy for describing the rabbit population, we might employ induction. We first prove that the statement works for the initial months (typically the first two months).
Then, assuming it holds for an arbitrary month \( n \), we demonstrate it also holds for month \( n+1 \). By connecting these two points, induction allows us to confidently say the sequence accurately models the rabbit problem for any month in the future.

Golden Ratio

The golden ratio, often symbolized by \( \phi \), is a special mathematical number approximately equal to 1.618033988. It's found by solving the equation \( L^2 = L + 1 \), which appears in the limit expression of ratios derived from Fibonacci numbers.
In the context of the Fibonacci sequence, as the sequence progresses, the ratio of successive Fibonacci numbers converges to the golden ratio. Mathematically expressing this, if \( a_n = \frac{f_{n+1}}{f_n} \), then over time, \( a_n \) approaches \( \phi \).
The golden ratio is remarkable for its widespread appearance not just in mathematics, but also in nature, art, and architecture. The aesthetic and structural appeal it offers is considered both mysterious and captivating. It is a key feature of self-similar patterns, enhancing our understanding of growth and design across various applications.