Problem 22 - Precalculus Mathematics for Calculus

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. Graph them to visualize the alternation.

Step 1: Understand the Sequence

The sequence given is \( a_n = 4 - 2(-1)^n \). This is a recursive sequence where the term depends on \( n \) being even or odd. Notably, \((-1)^n\) alternates between 1 and -1 when \( n \) is even and odd, respectively.

Step 2: Calculate Terms

We need to find the first 10 terms by substituting \( n = 1, 2, 3, \ldots, 10 \) into the sequence formula:- For \( n = 1, \ a_1 = 4 - 2(-1)^1 = 4 + 2 = 6 \)- For \( n = 2, \ a_2 = 4 - 2(-1)^2 = 4 - 2 = 2 \)- Continue this pattern for up to \( n = 10 \).The first 10 terms are: 6, 2, 6, 2, 6, 2, 6, 2, 6, 2.

Step 3: Graph the Sequence

Now, plot these terms using the graphing calculator. Use the x-axis to represent the term number (n) from 1 to 10 and the y-axis to represent the value of each term \( a_n \). You will see a pattern alternating between 6 and 2 for these values.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that allows you to visualize mathematical functions and sequences easily. It comes equipped with a large screen to display graphs and has the ability to handle complex calculations. For our exercise, it can help us graph the sequence defined by \( a_n = 4 - 2(-1)^n \).

To use a graphing calculator effectively, follow these steps:

• Enter the sequence formula into the calculator.
• Set the calculator to display the first 10 terms, which are 6, 2, 6, 2, 6, 2, 6, 2, 6, and 2.
• Plot these terms on the graph, using the term number \( n \) on the x-axis and the sequence value \( a_n \) on the y-axis.

This visual representation shows the alternating pattern of the sequence, making it easier for you to grasp the behavior and properties of such sequences.

Recursive Sequences

Recursive sequences are sequences where the next term is calculated based on one or more of the preceding terms. In our exercise, the sequence \( a_n = 4 - 2(-1)^n \) behaves differently depending on whether \( n \) is even or odd.

Here's a breakdown of how recursion works in the sequence:

• If \( n \) is odd, \((-1)^n = -1\), altering the sequence formula to \( a_n = 4 + 2 = 6 \).
• If \( n \) is even, \((-1)^n = 1\), changing the formula to \( a_n = 4 - 2 = 2 \).
• This creates an alternating pattern in the terms, contributing to the overall behavior of this particular recursive sequence.

Understanding recursive sequences is crucial as they are widely used in mathematics and computer science for defining patterns and processes.

Alternating Sequences

Alternating sequences, like the one in our example, involve terms that switch regularly between different values. The sequence \( a_n = 4 - 2(-1)^n \) is a perfect illustration of an alternating sequence as it alternates between the values of 6 and 2.

Key characteristics of alternating sequences:

• The alternation is driven by the component \((-1)^n\) in our formula, which swaps between -1 and 1 depending on the parity (odd or even) of \( n \).
• This results in a predictable pattern, making the sequence easy to analyze and understand.
• Graphically, this alternation produces a distinct zig-zag pattern, evident on the graph where even \( n \) produces a lower value and odd \( n \) a higher one.

Recognizing such patterns aids in predicting future terms and understanding the structure of sequences in mathematical contexts.