Problem 70 - 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 \(F_{m}\)

Step-by-Step Solution

Answer

The sequence \(G_{n}\) matches the Fibonacci sequence exactly for \(n = 1\) to 10.

Step 1: Understanding the Sequence Formula

The given sequence is defined by the formula: \[G_{n} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)\]This is a well-known formula that represents the Fibonacci sequence in closed form, known as Binet's formula.

Step 2: Using the Formula with a Calculator

To find the first 10 terms of the sequence, input the formula into a graphing calculator using the \([\text { TABLE }]\) function. Substitute values of \(n\) from 1 to 10 into the formula to compute each term. The sequence \(G_{n}\) should then be calculated and displayed.

Step 3: Calculating Each Term

Calculate the first few terms manually for verification:For \(n = 1\):\[G_{1} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{1}-(1-\sqrt{5})^{1}}{2^{1}}\right) = 1\]For \(n = 2\):\[G_{2} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{2}-(1-\sqrt{5})^{2}}{2^{2}}\right) = 1\]Continue similarly for terms up to \(n = 10\).

Step 4: Comparing with Fibonacci Sequence

The Fibonacci sequence \(F_{m}\) starts as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Compare this with the sequence \(G_{n}\) calculated using the formula. Notice the terms of \(G_{n}\) for \(n = 1\) to \(n = 10\) should match the corresponding Fibonacci numbers.

Key Concepts

Binet's formula

Binet's formula is a significant expression in mathematics as it provides a way to represent the Fibonacci sequence in what is known as a closed form. This means that you can find any term in the Fibonacci sequence without needing to calculate all the previous terms, which is typically required when using the recursive definition of the Fibonacci sequence. The formula is:\[ G_{n} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right) \]This remarkable formula involves irrational numbers like the square root of five, and cleverly uses these components to produce integer results that match Fibonacci numbers, which are natural numbers.To break it down further:

• \((1 + \sqrt{5})^n\) and \((1 - \sqrt{5})^n\) are powers of two conjugate roots, resulting from deriving Binet's formula from the characteristic equation of the Fibonacci sequence.
• Division by \(2^n\) helps normalize these roots into the sequence.
• Finally, dividing by \(\sqrt{5}\) scales the result to align with the Fibonacci sequence integers.

Binet's formula is an elegant mathematical approach that shows the power of using algebra and number theory to simplify sequence problems.

sequence formula

A sequence formula is a mathematical expression that allows us to find the terms of a sequence directly. Unlike recursive formulas, where each term depends on the previous ones, a sequence formula allows us to compute any term directly using its position in the sequence. In the context of the exercise, the sequence formula for the Fibonacci sequence, known as Binet's formula, is particularly useful for:

• Calculating large Fibonacci numbers quickly, without the need to know the entire sequence beforehand.
• Demonstrating the linkages between algebraic expressions and number patterns, aiding in deeper understanding of mathematical properties.

Understanding sequence formulas broadens our comprehension of various mathematical sequences. These formulas often involve sums, differences, or powers that illustrate how sequences grow and behave. By mastering sequence formulas, students can tackle a wider range of problems with greater efficiency.

graphing calculator

Graphing calculators are powerful tools in modern mathematics education, enabling students to visualize and compute complex equations conveniently. Utilizing a graphing calculator for calculating the sequence defined by Binet's formula can greatly enhance understanding.### How to Use a Graphing Calculator for Sequences1. **Input the Formula**: - Enter Binet's formula into the calculator. Make sure to represent all parts of the formula accurately, including parentheses and division lines.2. **Use the TABLE Function**: - This feature allows you to calculate multiple terms at once by setting up a table where you input the starting and ending values of \(n\).3. **Analyze and Compare**: - Once you have the first 10 terms, compare them to the expected values from the Fibonacci sequence. This can be done using either the built-in features or additional computations.Graphing calculators not only assist in computing sequences but also boost students' confidence in verifying manual calculations. They allow for strong visual representation, fostering a deeper understanding of the mathematics involved.