Problem 70 - Precalculus Mathematics for Calculus

Question

Define the sequence $$ G_{n}=\frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right) $$ Use the \([\text { ABLE }]\) command on a graphing calculator to find the first 10 terms of this sequence. Compare to the Fibonacci sequence \(F_{n}\)

Step-by-Step Solution

Answer

The first 10 terms of \( G_n \) match the Fibonacci sequence.

Step 1: Understand the Formula

The sequence is given by the formula \( G_{n} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right) \). This formula is known as "Binet's formula" and is used to calculate the Fibonacci numbers directly.

Step 2: Recall the Fibonacci Sequence

The Fibonacci sequence \( F_{n} \) starts with \( F_0 = 0 \) and \( F_1 = 1 \). For \( n \geq 2 \), each term is the sum of the two preceding ones: \( F_{n} = F_{n-1} + F_{n-2} \). The sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

Step 3: Use Graphing Calculator Table Function

Enter \( G_n = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right) \) into a graphing calculator to generate the sequence. Use the table function to compute the first 10 terms, starting from \( n = 0 \).

Step 4: Calculate the First 10 Terms of \( G_n \)

Using the graphing calculator, the first 10 terms of \( G_n \) are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. These are computed values derived from Binet's formula.

Step 5: Compare with the Fibonacci Sequence

Compare the first 10 terms of the generated sequence \( G_n \) with the Fibonacci sequence \( F_n \). Both sequences are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, confirming that \( G_n \) indeed matches the Fibonacci sequence.

Key Concepts

Binet's Formula

Binet's formula is a fascinating mathematical expression that allows us to calculate Fibonacci numbers directly without having to compute every preceding term. The formula is:\[ G_{n} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right) \]This elegant formula was discovered by Jacques Philippe Marie Binet and provides a closed-form solution to the classic Fibonacci sequence.

• The formula incorporates the golden ratio, which appears as approximately 1.61803 in its expression.
• The terms \( (1+\sqrt{5})^{n} \) and \( (1-\sqrt{5})^{n} \) are known as conjugates in this context.
• This expression applies the characteristics of the golden ratio to find values in a growing sequence that obeys the Fibonacci property.

Understanding and utilizing Binet's formula is a great way to appreciate the connection between sequences and algebra, as it simplifies the process of finding Fibonacci numbers, especially for larger \( n \).

Mathematical Sequences

Mathematical sequences, like the Fibonacci sequence, are ordered lists of numbers that follow a specific rule or pattern. The Fibonacci sequence is particularly famous due to its simple rule.In the Fibonacci sequence, each term is the sum of the two preceding numbers, beginning with 0 and 1:

• \( F_0 = 0 \)
• \( F_1 = 1 \)
• \( F_2 = F_1 + F_0 = 1\)
• \( F_3 = F_2 + F_1 = 2\)
• ...continuing indefinitely

These kinds of sequences are not only essential in mathematics but also broadly applied in science, computer algorithms, and finance due to their inherent predictability and structured growth patterns. Recognizing and understanding these sequences helps in modeling natural phenomena and solving practical problems.

Graphing Calculator Usage

Using a graphing calculator can greatly enhance your ability to explore mathematical sequences like the Fibonacci numbers. Here's a simple guide to using a graphing calculator for finding terms of a sequence:1. **Enter the Formula**: Start by entering the specific formula you want to explore. For Binet's formula, input: \[ \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right) \]2. **Use the Table Function**: This feature allows you to compute values easily for a range of \( n \) values. It's particularly useful for sequences.3. **Generate Terms**: Set the calculator to compute terms from the desired starting point, like \( n = 0 \), and see the first 10 terms mirrored against expected results.4. **Compare Output**: Once you have the computed terms, you can compare them with the known data, verifying results or discovering patterns.By mastering the use of a graphing calculator, you simplify the process of finding sequence terms and verifying their correctness. It becomes an indispensable tool for mathematical exploration and learning.