Problem 20 - Algebra and Trigonometry

Question

Use a graphing calculator to do the following. (a) Find the first 10 terms of the sequence. (b) Graph the first 10 terms of the sequence. \(a_{n}=4-2(-1)^{n}\)

Step-by-Step Solution

Answer

The first 10 terms are: 6, 2, 6, 2, 6, 2, 6, 2, 6, 2.

Step 1: Understand the Sequence

The given sequence is defined as \( a_n = 4 - 2(-1)^n \). This means that the sequence alternates based on the value of \((-1)^n\). For even \(n\), \((-1)^n\) is 1, and for odd \(n\), \((-1)^n\) is -1.

Step 2: Calculate the First 10 Terms

To find the first 10 terms, we compute \(a_n\) for \( n = 1 \) to \( 10 \). 1. \(a_1 = 4 - 2(-1)^1 = 4 + 2 = 6\) 2. \(a_2 = 4 - 2(-1)^2 = 4 - 2 = 2\) 3. \(a_3 = 4 - 2(-1)^3 = 4 + 2 = 6\) 4. \(a_4 = 4 - 2(-1)^4 = 4 - 2 = 2\) 5. \(a_5 = 4 - 2(-1)^5 = 4 + 2 = 6\) 6. \(a_6 = 4 - 2(-1)^6 = 4 - 2 = 2\) 7. \(a_7 = 4 - 2(-1)^7 = 4 + 2 = 6\) 8. \(a_8 = 4 - 2(-1)^8 = 4 - 2 = 2\) 9. \(a_9 = 4 - 2(-1)^9 = 4 + 2 = 6\) 10. \(a_{10} = 4 - 2(-1)^{10} = 4 - 2 = 2\). The first 10 terms are: 6, 2, 6, 2, 6, 2, 6, 2, 6, 2.

Step 3: Graph the Sequence

Use your graphing calculator to plot the first 10 terms of the sequence. Set the x-axis to represent \( n \) from 1 to 10, and the y-axis to represent the sequence terms \( a_n \). You should observe two horizontal lines, one at \( a_n = 6 \) for odd \( n \) and another at \( a_n = 2 \) for even \( n \), alternating between these two values as \( n \) increases.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can help visualize mathematical concepts, particularly sequences. Unlike basic calculators, graphing calculators have the ability to plot points and connect them to form graphs.

To use a graphing calculator for sequences, you'll first need to enter the formula of the sequence into the calculator. For example, if given the sequence \(a_n = 4 - 2(-1)^n\), you can input this equation into the calculator, specifying the range of \(n\) from 1 to your desired number of terms, like the first 10 terms.

• Set up the sequence function on the graphing display.
• Define the range for \(n\), typically starting at 1 and ending as needed.
• Observe the plotted points, which represent the sequence terms.

The visual representation on a graph can often make patterns easier to recognize, aiding in the understanding of sequence behaviors.

Alternating Sequence

An alternating sequence is a sequence of numbers that switches between two or more values in a regular pattern.
An example of this is the sequence defined by the formula: \(a_n = 4 - 2(-1)^n\). This sequence alternates between two values as \((-1)^n\) changes its sign based on whether \(n\) is odd or even:

• For even \(n\) (like 2, 4, 6, etc.), \((-1)^n\) equals \(1\), making the sequence term \(a_n = 2\).
• For odd \(n\) (like 1, 3, 5, etc.), \((-1)^n\) equals \(-1\), resulting in \(a_n = 6\).

This creates a clear, easily recognizable pattern alternating between these two values. Such sequences are useful in various mathematical and physical applications where cyclic behaviors occur.

Graphing Sequences

Graphing sequences can enhance your understanding by providing a visual representation of the sequence's behavior. When plotting the sequence \(a_n = 4 - 2(-1)^n\), you'll notice certain characteristics:

• The x-axis typically represents the term number \(n\).
• The y-axis shows the value of each sequence term \(a_n\).

For the sequence provided, the graph will display horizontal lines at \(a_n = 6\) for odd \(n\), and \(a_n = 2\) for even \(n\), clearly illustrating the alternating nature.

By plotting these terms on a graphing calculator, it becomes evident how the sequence "alternates" through a simple visual inspection, making complex numerical patterns accessible.