Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

The Fibonacci sequence The Fibonacci sequence is defined recursively by $$ a_{1}=1, \quad a_{2}=1, \quad a_{k+1}=a_{k}+a_{k-1} \text { for } k \geq 2 . $$ (a) Find the first ten terms of the sequence. (b) The terms of the sequence \(r_{k}=a_{k+1} / a_{k}\) give progressively better approximations to \(\tau\), the golden ratio. Approximate the first ten terms of this sequence.

Step-by-Step Solution

Answer

The first ten Fibonacci numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. The ratios \( r_k \) are: 1, 2, 1.5, 1.6667, 1.6, 1.625, 1.6154, 1.6190, 1.6176, approaching \( \tau \).

Step 1: Initial Assignment of Fibonacci Terms

The Fibonacci sequence is given with its initial terms as \( a_1 = 1 \) and \( a_2 = 1 \). These will be the first two terms. We need to determine the next eight terms based on the recursive formula \( a_{k+1} = a_k + a_{k-1} \) for \( k \geq 2 \).

Step 2: Calculate the Next Eight Fibonacci Terms

Starting from the third term, use the recursive formula to find each next term:- \( a_3 = a_2 + a_1 = 1 + 1 = 2 \).- \( a_4 = a_3 + a_2 = 2 + 1 = 3 \).- \( a_5 = a_4 + a_3 = 3 + 2 = 5 \).- \( a_6 = a_5 + a_4 = 5 + 3 = 8 \).- \( a_7 = a_6 + a_5 = 8 + 5 = 13 \).- \( a_8 = a_7 + a_6 = 13 + 8 = 21 \).- \( a_9 = a_8 + a_7 = 21 + 13 = 34 \).- \( a_{10} = a_9 + a_8 = 34 + 21 = 55 \).

Step 3: First Ten Terms of the Fibonacci Sequence

The first ten terms of the Fibonacci sequence are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Step 4: Define and Initialize the Ratio Sequence \( r_k \)

The sequence \( r_k = \frac{a_{k+1}}{a_k} \) provides a ratio between consecutive Fibonacci numbers, approximating the golden ratio \( \tau \). We will calculate this ratio for the first nine necessary terms, as \( r_k \) uses terms starting from \( a_2 \) and goes up to \( a_{10} \).

Step 5: Calculate the Ratio Sequence Terms

Calculate each ratio:- \( r_1 = \frac{a_2}{a_1} = \frac{1}{1} = 1 \).- \( r_2 = \frac{a_3}{a_2} = \frac{2}{1} = 2 \).- \( r_3 = \frac{a_4}{a_3} = \frac{3}{2} \approx 1.5 \).- \( r_4 = \frac{a_5}{a_4} = \frac{5}{3} \approx 1.6667 \).- \( r_5 = \frac{a_6}{a_5} = \frac{8}{5} \approx 1.6 \).- \( r_6 = \frac{a_7}{a_6} = \frac{13}{8} \approx 1.625 \).- \( r_7 = \frac{a_8}{a_7} = \frac{21}{13} \approx 1.6154 \).- \( r_8 = \frac{a_9}{a_8} = \frac{34}{21} \approx 1.6190 \).- \( r_9 = \frac{a_{10}}{a_9} = \frac{55}{34} \approx 1.6176 \).

Step 6: Summary of the Ratio Sequence

The first nine terms of the sequence \( r_k \) are: 1, 2, 1.5, 1.6667, 1.6, 1.625, 1.6154, 1.6190, 1.6176, providing approximate values that get closer to the golden ratio \( \tau \).

Key Concepts

Golden Ratio

The golden ratio is a fascinating mathematical concept, often denoted by the Greek letter \( \tau \) (or sometimes \( \phi \)). It approximately equals 1.6180339887\( \ldots \) and is known for its unique properties and aesthetic appeal.
Many natural phenomena and works of art follow this ratio, including the spiral of seashells and the structure of galaxies.
The golden ratio emerges through the Fibonacci sequence, where the ratio of successive Fibonacci numbers converges to this magical number.
As we calculate more terms in the Fibonacci sequence and take the ratio of consecutive numbers, these values get closer and closer to the golden ratio. This makes the sequence a perfect illustration of how mathematical concepts can be intertwined with nature and art.

Recursive Formula

A recursive formula is used to generate the terms of a sequence where each term is defined in relation to one or more previous terms.
In the Fibonacci sequence, each term beyond the first two is the sum of the two preceding terms.
This can be represented by the recursive formula:

• \( a_1 = 1 \)
• \( a_2 = 1 \)
• For \( k \geq 2 \), \( a_{k+1} = a_k + a_{k-1} \)

This simple formula uniquely defines the entire sequence.
By understanding the recursive nature, you can easily continue the sequence for as many terms as needed, simply using the defined operation.
Recursive formulas highlight the beautifully uncomplicated yet powerful patterns that can be found in mathematics.

Fibonacci Ratio

The Fibonacci ratio refers to the ratio between successive terms in the Fibonacci sequence, which aids in reaching the golden ratio.
Denoted as \( r_k = \frac{a_{k+1}}{a_k} \), these ratios start as simple fractions, such as \( 1/1 \), \( 2/1 \), and \( 3/2 \).
As you calculate more ratios, they approximate closer to the golden ratio, showcasing an incredible numerical convergence.
The Fibonacci ratio is important because it visualizes the idea of approaching a specific numerical target through a series of rational steps.

Sequence Approximation

Sequence approximation involves using a sequence to estimate or approach a particular value.
In the case of the Fibonacci sequence, the goal is to approach the golden ratio using the Fibonacci ratio.
As the sequence progresses, the ratios between consecutive Fibonacci numbers become a closer approximation of the golden ratio.
This approach is insightful, showing how simple numerical operations can reveal complex mathematical truths.
Through approximation, we see how iterative processes can be used to come ever closer to a chosen target, an essential concept in mathematics.