Question

$$\begin{aligned}&\text { The following expressions are called continued fractions: }\\\&1+\frac{1}{x} \quad 1+\frac{1}{1+\frac{1}{x}} \quad 1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}} \ldots\end{aligned}$$$$\text { Each simplifies to an expression of the form } \frac{a x+b}{b x+c}$$Trace the successive values of \(a, b,\) and \(c\) as you "continue" the fraction. Can you discover the pattern of these values? Go to the library and research Fibonacci numbers. Write a report on your findings.

Short answer

The values of a, b, and c in the continued fraction follow the Fibonacci sequence.

Step-by-step solution

- Simplify the first continued fraction

Start with the simplest form of the given continued fraction: \[ 1 + \frac{1}{x} \]This can be written as: \[ \frac{x + 1}{x} \]Setting this equal to \( \frac{a x + b}{b x + c} \), we can identify the values of \(a, b,\) and \(c\): \( a = 1, b = 1, c = 0 \).

- Simplify the second continued fraction

Next, simplify the second continued fraction: \[ 1 + \frac{1}{1 + \frac{1}{x}} \]Start by simplifying the inner expression: \[ 1 + \frac{1}{x} = \frac{x + 1}{x} \]Now substitute back: \[ 1 + \frac{1}{\frac{x + 1}{x}} = 1 + \frac{x}{x + 1} = \frac{x + 1 + x}{x + 1} = \frac{2x + 1}{x + 1} \]So, \( a = 2, b = 1, c = 1 \).

- Simplify the third continued fraction

Next, simplify the third continued fraction: \[ 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} \]Start by simplifying the innermost expression:\[ 1 + \frac{1}{1 + \frac{1}{x}} = \frac{2x + 1}{x + 1} \]Now substitute back:\[ 1 + \frac{1}{\frac{2x + 1}{x + 1}} = 1 + \frac{x + 1}{2x + 1} = \frac{2x + 1 + x + 1}{2x + 1} = \frac{3x + 2}{2x + 1} \]So, \( a = 3, b = 2, c = 1 \).

- Identify the pattern

Observe the sequence of \( a, b, \) and \( c \): \[ a: 1, 2, 3, \ldots \]\[ b: 1, 1, 2, \ldots \]\[ c: 0, 1, 1, \ldots \]The numerators and denominators follow the Fibonacci sequence.

- Research Fibonacci Numbers

Research the Fibonacci sequence. Each term in the Fibonacci sequence is the sum of the two preceding ones, starting from 0 and 1. The first few terms are: \(0, 1, 1, 2, 3, 5, 8, \ldots \)

- Write a report

Write a report detailing how the values of \( a, b, \) and \( c \) in the continued fraction follow the Fibonacci sequence, providing examples and mathematical reasoning.

Key concepts

Fibonacci Sequence

The Fibonacci sequence is a fascinating series of numbers that has captivated mathematicians for centuries. It starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. Formally, the sequence is defined as:

• F(0) = 0
• F(1) = 1
• F(n) = F(n-1) + F(n-2) for n > 1

So, the first few numbers in this series are: 0, 1, 1, 2, 3, 5, 8, 13, and so on.
The Fibonacci sequence appears in many areas of mathematics and science, from the branching patterns of trees to the arrangement of leaves on a stem.

In the context of evaluating continued fractions, we see a recurring pattern of values that match the Fibonacci sequence. As you simplify these fractions, the numerators and denominators align with the Fibonacci numbers, demonstrating the wide-reaching influence of this sequence.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials.
These types of expressions are fundamental in algebra, paving the way for understanding more complex operations in mathematics.
Continued fractions, like those in the exercise, often simplify to rational expressions. For example:

• The fraction \(1 + \frac{1}{x}\) simplifies to \( \frac{x+1}{x} \).

Every continued fraction can be expressed in the form \( \frac{a x + b}{b x + c} \), where a, b, and c are constants determined through the steps of simplification. These constants follow patterns that can often be traced back to well-known sequences like the Fibonacci sequence.
The importance of rational expressions extends beyond just algebra—they are critical in calculus, number theory, and many applied mathematics fields, making them a core concept to master.

Pattern Recognition

Pattern recognition is the ability to observe patterns and regularities in data. It's an essential skill in mathematics, science, and everyday problem-solving.
When you simplify continued fractions, observing the patterns in the values of a, b, and c can help you predict subsequent results without doing all of the tedious calculations.
For instance, in the exercise, you found that:

• a: 1, 2, 3, ...
• b: 1, 1, 2, ...
• c: 0, 1, 1, ...

This pattern forms a sequence resembling the Fibonacci series. In mathematics, encountering such patterns can lead to generalizations and deeper insights, making it easier to solve similar problems in the future.
Developing your pattern recognition skills will help you not only with specific exercises but also with broader mathematical concepts and real-life applications.