Problem 95 - Essential Calculus

Question

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\) (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by \(b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1\) (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho .\) Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).

Step-by-Step Solution

Answer

The first 12 terms of the Fibonacci sequence are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The first 10 terms of the sequence \(b_{n}=\frac{a_{n+1}}{a_{n}}\) are approximately: 1, 2, 1.5, 1.6667, 1.6, 1.625, 1.6154, 1.6190, 1.6176, 1.6182. The golden ratio \(\rho = (1 + \sqrt{5}) / 2\) is approximately 1.61803.

Step 1: Construct the Fibonacci Sequence

Start with 1 and 1 as the first two terms. For each additional term, add the two preceding terms together. The first 12 terms of the Fibonacci sequence using the defined recursive function \(a_{n+2}=a_{n}+a_{n+1}\) are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

Step 2: Construct the Sequence \(b_{n}=\frac{a_{n+1}}{a_{n}}\)

The new sequence uses the values from the Fibonacci sequence. Use the first 11 terms of the Fibonacci sequence generated to create the first 10 terms of new sequence \(b_{n}\), they are: 1, 2, 1.5, 1.6667, 1.6, 1.625, 1.6154, 1.6190, 1.6176, 1.6182.

Step 3: Prove the Relationship \(b_{n}=1+\frac{1}{b_{n-1}}\)

Rearrange the recursive Fibonacci formula: \(a_{n+2}=a_{n}+a_{n+1}\) in terms of \(a_{n+1}\) and \(a_{n}\) we get: \(a_{n+2}/a_{n+1} = a_{n}/a_{n+1} + 1\), which equates to \(b_{n+1} = b_{n}+1/b_{n}\).

Step 4: Define and Solve for the Golden Ratio

The limit as \(n\) approaches infinity of the sequence \(b_n\) is defined as the golden ratio \(\rho = \lim_{n \to \infty} b_{n}\). Since the limit of \(b_n\) also satisfies the relationship \(b_n = 1 + 1/b_{n-1}\), we can substitute \(\rho\) to obtain \(\rho = 1 + 1 / \rho\), which simplifies to \(\rho² - \rho - 1 = 0\). Solving this quadratic equation gives the solutions \(\rho = (1 ± \sqrt{5}) / 2\). Since \(\rho\) is a ratio and cannot be negative, we reject the negative root, and thus \(\rho = (1 + \sqrt{5}) / 2\) which is approximately equal to 1.61803.

Key Concepts

Golden Ratio

The golden ratio, often denoted by the Greek letter \( \rho \), is a special number approximately equal to 1.61803. This unique ratio has fascinated mathematicians, artists, and architects for centuries, appearing in various aspects of art, architecture, and nature. It can be defined mathematically as the limit of the ratio of successive Fibonacci numbers, \( \lim_{n \rightarrow \infty} b_n = \rho \). This implies that as we progress through the Fibonacci sequence, the quotient of consecutive terms increasingly approximates \( \rho \).
To understand why \( \rho = 1 + \frac{1}{\rho} \), consider the equation: \( b_n = 1 + \frac{1}{b_{n-1}} \). If we assume that \( b_n \) reaches a constant value \( \rho \) as \( n \) approaches infinity, the formula simplifies to \( \rho = 1 + \frac{1}{\rho} \). Solving this equation through quadratic techniques results in \( \rho^2 - \rho - 1 = 0 \). The positive solution to this equation is \( \rho = \frac{1 + \sqrt{5}}{2} \), illustrating its special significance in mathematics.

Recursive Sequences

Recursive sequences are a fundamental concept in mathematics, where each term in the sequence is defined based on one or more of its preceding terms. The Fibonacci sequence is a classic example where the relation \( a_{n+2} = a_n + a_{n+1} \) describes how each term is the sum of the previous two.
Using recursion allows for the construction of sequences without having to rely on explicit formulas for each term. Instead, you start with a known set of initial terms and use the recursive relation to build the sequence.

• Initial terms are crucial because they provide the necessary starting point for the recursive process.
• Understanding recursion helps in breaking down complex problems into smaller, more manageable components.
• This approach can often reveal hidden patterns or properties inherent in the sequence.

Recursive sequences are powerful in fields like computer science and economics, where iterative processes and complex systems often benefit from recursion.

Quadratic Equations

Quadratic equations feature prominently in various areas of mathematics and science. They have the general form \( ax^2 + bx + c = 0 \). In the context of the Fibonacci sequence and the golden ratio, they emerge naturally when solving equations involving recursive sequences.
The connection between the golden ratio and quadratic equations is apparent when we rearrange \( \rho = 1 + \frac{1}{\rho} \) to quadratic form: \( \rho^2 - \rho - 1 = 0 \). Solving this using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = -1 \), gives us solutions for \( \rho \).

• The discriminant \( b^2 - 4ac \) determines the nature of the roots, with a positive value indicating real and distinct roots.
• For the golden ratio, \( b^2 - 4ac = 1 + 4 = 5 \), allowing the calculation of \( \frac{1 + \sqrt{5}}{2} \).
• Understanding quadratic equations is essential for solving problems involving parabolas, trajectory analysis, and many geometry problems.

Quadratic equations provide a structural approach for analyzing patterns and relationships in numerical sequences like the Fibonacci numbers.