Problem 68 - Algebra and Trigonometry

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 \([\text { TABLE }]\) 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 sequence \( G_n \) is exactly the Fibonacci sequence.

Step 1: Understand the Formula

The given sequence is defined as \( G_{n} = \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^{n} - (1-\sqrt{5})^{n}}{2^{n}} \right) \). This resembles Binet's formula for the Fibonacci sequence, suggesting that \( G_{n} \) might produce Fibonacci numbers.

Step 2: Set Up the Calculator

Enter the expression \( \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^{n} - (1-\sqrt{5})^{n}}{2^{n}} \right) \) into the calculator. Use the \("[ \text{TABLE} ]"\) function to automate the computation of the first 10 terms for \( n = 0 \) to \( n = 9 \).

Step 3: Calculate the First Term

For \( n = 0 \), compute \( G_{0} \) as follows: \( G_{0} = \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^{0} - (1-\sqrt{5})^{0}}{2^{0}} \right) = \frac{1}{\sqrt{5}}(1 - 1) = 0 \).

Step 4: Calculate the Second Term

For \( n = 1 \), compute \( G_{1} \) as follows: \( G_{1} = \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^{1} - (1-\sqrt{5})^{1}}{2^{1}} \right) = 1 \).

Step 5: Calculate the Third Term

For \( n = 2 \), compute \( G_{2} \) as follows: \( G_{2} = \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^{2} - (1-\sqrt{5})^{2}}{2^{2}} \right) = 1 \).

Step 6: Calculate Additional Terms

Carry on this process until \( n = 9 \). You'll find that \( G_{3} = 2 \), \( G_{4} = 3 \), \( G_{5} = 5 \), \( G_{6} = 8 \), \( G_{7} = 13 \), \( G_{8} = 21 \), and \( G_{9} = 34 \).

Step 7: Compare to the Fibonacci Sequence

As observed, the terms \( G_0 \) through \( G_9 \) are precisely the first ten Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. This confirms \( G_n \) produces the Fibonacci sequence by using Binet's formula.

Key Concepts

Binet's Formula

Binet's formula is a mathematical expression that provides a way to directly calculate the n-th term in the Fibonacci sequence without having to step through each preceding term. This is a handy tool because it gives a neat and concise representation of these famous numbers. Binet's formula especially comes in handy when you need to find larger terms in the sequence. This formula is given as:

• \( F_n = \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^{n} - (1-\sqrt{5})^{n}}{2^{n}} \right) \)

This formula uses the golden ratio, \(\phi = \frac{1+\sqrt{5}}{2}\), and its conjugate, \(1-\phi\). It shows off the deep connection of the Fibonacci numbers with the golden ratio, which appears throughout nature and mathematics. When applying this formula, you can effortlessly compute any Fibonacci number by plugging in your desired value of \(n\).

Sequence Calculation

Calculating a sequence involves determining a set of numbers one by one. For example, if you want to find initial terms using Binet's formula, you'd start with your smallest \( n \), which is typically 0 or 1, and compute that term. Then, you'd progress to the next term by increasing \( n \) by one and repeating the calculation. This sequential process is quite similar to following any step-by-step procedure. However, the sequence calculation, when dealing with a formula as complex as Binet's, can be intensive. Thankfully, with Binet's formula, the calculations lead to Fibonacci numbers directly. This was demonstrated in our example where the terms \(G_0\) to \(G_9\) were calculated using the formula, illustrating a method to produce Fibonacci numbers without iterative summation of previous terms.

Graphing Calculator Usage

Graphing calculators are terrific tools for exploring sequences such as those defined by Binet's formula. To efficiently calculate terms of the sequence, you can use the "[ TABLE ]" function. This feature automates tasks that would otherwise require repeated manual entry of numbers and operations.

Here's how to do it:

• Input the expression \( \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^{n} - (1-\sqrt{5})^{n}}{2^{n}} \right) \) into your calculator.
• Select the "[ TABLE ]" option.
• You can then set values of \( n \) ranging from 0 to the desired number to automatically populate the table with corresponding sequence terms.

This approach not only saves time but also minimizes the chance of arithmetic errors, allowing for a focus on observing patterns and understanding the sequence's behavior.

Mathematical Sequences

A mathematical sequence is an ordered list of numbers that often follows a specific pattern or rule. The Fibonacci sequence, which is generated by adding the two preceding numbers, is a well-known example. This sequence begins as 0, 1, 1, 2, 3, 5, 8, and so on, each subsequent number being the sum of the previous two. Such sequences are fundamental in math and appear in various applications, from computer algorithms to biological settings.

The beauty of sequences is typically shown by their simplicity and predictability. Sequences not only fascinate with their defining rules but also serve as a model for complex systems in the natural world. By understanding these fundamental patterns, students are better prepared to tackle complexity in advanced mathematical and scientific studies.