Problem 37 - Precalculus with Limits

Question

In Exercises 37-42, use a graphing utility to graph the first 10 terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = \dfrac{2}{3} n \)

Step-by-Step Solution

Answer

The first 10 terms of the sequence when plotted form a straight line with a positive slope. The tenth term of the sequence is \( \dfrac{20}{3} \).

Step 1: Identification

Identify the sequence presented: \( a_n = \dfrac{2}{3} n \). The task is to plot this on a graph.

Step 2: Calculation of sequence values

To find the first 10 terms of the sequence, plug \( n \) = 1, 2, 3, ..., 10 into the formula. The tenth term (i.e., when \( n = 10 \)) gives \( a_{10} = \dfrac{2}{3} \times 10 = \dfrac{20}{3} \).

Step 3: Listing sequence values

Now list these values in a table like format. Keeping in mind that the \( n \)th term's \( x \) - axis coordinate is \( n \) and the \( y \) - axis coordinate is \( a_n \). The first three sequence values are for example: (1, \( \dfrac{2}{3} \)), (2, \( \dfrac{4}{3} \)), (3, 2).

Step 4: Plotting the sequence

Using the table listed in the previous step, plot the points on a graphing utility. The x-coordinate represents the term number and the y-coordinate is the value of the term. Connect the points to visualize the pattern created by the sequence.

Key Concepts

Arithmetic Sequences

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which each term after the first is obtained by adding a constant difference, called the common difference, to the previous term. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3 since each term increments by 3. To determine if a sequence is arithmetic, you can subtract any term from the one that follows it. If the difference is constant, you're dealing with an arithmetic sequence.

An arithmetic sequence can be expressed using the formula:
\[a_n = a_1 + (n-1)d\]
where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference. For the exercise provided, the sequence given by \(a_n = \dfrac{2}{3}n\) is not a classic arithmetic sequence because the difference between terms isn't consistent; instead, it increases proportionally with \(n\).

Sequence Graphing Utility

A sequence graphing utility is a tool often used in mathematics to visualize sequences. By plotting a sequence on a graph, we can easily see patterns, behaviors, and the overall direction of the sequence. Most graphing utilities can handle different types of sequences and are able to accommodate a wide range of mathematical functions. Students can use these utilities to input a sequence formula, define the range of terms to be displayed, and the tool will automatically generate the points representing terms of the sequence on a coordinate grid.

Such utilities are particularly helpful when exploring complex sequences that might not display obvious patterns at first glance. In the context of the given exercise, using a graphing utility helps students visualize how the values of the sequence \(a_n = \dfrac{2}{3}n\) evolve as \(n\) increases.

Calculating Sequence Values

Calculating the values of a sequence is a foundational skill in mathematics, especially when you are looking at patterns or trying to identify the type of sequence. For a sequence defined by a specific formula, like \(a_n = \dfrac{2}{3}n\), you calculate individual terms by substituting the term number for \(n\) in the formula. This process is known as term evaluation.

For example, to find the 5th term \(a_5\) in the sequence mentioned, you substitute 5 for \(n\):\[a_5 = \dfrac{2}{3} \times 5 = \dfrac{10}{3}\]
By calculating the first several terms, patterns may become apparent which can help in understanding the sequence's behavior more comprehensively.

Graphing Sequence Terms

Graphing the terms of a sequence can be illuminating for students as it transforms abstract numbers into a visual pattern. To graph sequence terms, treat the term's index number as the x-coordinate and the term's value as the y-coordinate. For the sequence provided, \(a_n = \dfrac{2}{3}n\), each point you plot on the graph will have coordinates \((n, \dfrac{2}{3}n)\).

Begin by plotting the first term at \((1, \dfrac{2}{3})\), the second term at \((2, \dfrac{4}{3})\), and so on until you reach the desired number of terms. After plotting, you can connect the dots to visualize the trend of the sequence. This visual representation helps in understanding how quickly the sequence increases or decreases as the term number grows.