Problem 24 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

First ten terms: 6, 2, 6, 2, 6, 2, 6, 2, 6, 2; Graph shows alternating pattern.

Step 1: Evaluate the Formula for Each Term

The formula for the sequence is given as \(a_n = 4 - 2(-1)^n\). We need to evaluate this formula for \(n = 1, 2, ..., 10\). - 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 calculating for \(n = 3\) through \(n = 10\).

Step 2: List the First Ten Terms

After evaluating the formula for each \(n\), we have:- \(a_3 = 6\), \(a_4 = 2\)- \(a_5 = 6\), \(a_6 = 2\)- \(a_7 = 6\), \(a_8 = 2\)- \(a_9 = 6\), \(a_{10} = 2\) Thus, the first ten terms are: 6, 2, 6, 2, 6, 2, 6, 2, 6, 2.

Step 3: Graph the Terms

Using a graphing calculator, enter each term of the sequence as discrete points. Plot the first ten terms: (1,6), (2,2), (3,6), (4,2), (5,6), (6,2), (7,6), (8,2), (9,6), (10,2), observing the alternating pattern shown by these terms.

Key Concepts

Graphing Calculator

A graphing calculator is a handy tool that allows you to visualize sequences and equations. Unlike basic calculators, it can plot graphs and handle complex functions. To graph a sequence, you'll generally be inputting data points derived from an equation, just like we have here with the sequence formula.

Here's a simple way to plot the terms using a graphing calculator:

• Start by entering your formula for the sequence: for example, the formula given as \(a_n = 4 - 2(-1)^n\).
• Calculate each term individually, from \(n = 1\) to \(n = 10\), by substituting the values of \(n\).
• Input these calculated values as discrete points. A discrete point refers to individual data entries rather than a continuous line.
• Finally, use the graph feature to plot these points. You'll see each term as a separate point on your graph, illustrating their distribution and pattern.

Always check your graph to make sure it accurately represents the data you've inputted.

Alternating Sequence

An alternating sequence is a type of sequence in which the sign or value of the terms reverses in some regular pattern. With the provided sequence formula \(a_n = 4 - 2(-1)^n\), the pattern is evident due to the term \((-1)^n\), which alternates between positive and negative values as \(n\) increases.

Here's how it works:

• For odd \(n\), \((-1)^n\) equals \(-1\), making the sequence term \(a_n = 4 + 2 = 6\).
• For even \(n\), \((-1)^n\) equals \(1\), making the sequence term \(a_n = 4 - 2 = 2\).

The sequence alternates between 6 and 2, leading to the ten terms: 6, 2, 6, 2, 6, 2, 6, 2, 6, 2. Such patterns are helpful in identifying sequences quickly and solving similar problems efficiently.

Recognizing an alternating sequence can make it easier to understand its behavior over a series of terms.

Discrete Points

Discrete points refer to data that can be distinctly separated and graphed individually, rather than being part of a continuous curve or line. In the context of sequences, each term is represented as a point on a graph, with its position determined by the term number \(n\) and the value \(a_n\).

To understand how to represent a sequence with discrete points:

• Consider each term as a point on a coordinate plane, with \(n\) as the x-coordinate and \(a_n\) as the y-coordinate.
• For this particular sequence, the points plotted would be \((1,6), (2,2), (3,6),\) etc., according to the calculated values for each \(n\).
• Discrete graphs help you see individual term differences, making patterns, such as the alternation seen in this sequence, easier to recognize.
• These graphs do not connect points with lines, emphasizing the individuality of each term.

By graphing discrete points, the pattern of the sequence becomes visually evident, which aids in both comprehension and analysis.