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Chapter 9: Problem 21

Use a graphing utility to graph the first 10 terms of the sequence. $$ a_{n}=\frac{2}{3} n $$

Short Answer

Expert verified
The first 10 terms of the sequence are: \( \frac{2}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{10}{3}, 4, \frac{14}{3}, \frac{16}{3}, 6, \frac{20}{3} \). Once generated, the graph should reflect these values and show a linear pattern with a positive slope, indicating a steadily increasing sequence.

Step by step solution

01

Understand the Sequence

First, we need to understand the sequence. Here, \( a_{n} \) is an arithmetic sequence defined algorithmically by the formula \( a_{n} = \frac{2}{3} n \), where \( n \) is the term number. This means that to find a given term in the sequence, we substitute the term's position number for \( n \) in the formula.
02

Determine The First 10 Terms

Next, we need to find the first 10 terms of the sequence. To do this, we will substitute the values \( n = 1, n = 2, ..., n = 10 \) into the formula and carry out the calculation. As given, \( a_{n} = \frac{2}{3} * n \), substituting \( n = 1, 2, 3, ..., 10 \) will give first 10 terms of the sequence.
03

Graphing The Sequence

In this step, we can use a graphing utility to plot these values. For example, on the x-axis we could place the term numbers (1, 2, ..., 10) and on the y-axis the corresponding sequence values. Each term will then appear as a point in the coordinate system, generating a discrete graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
A graphing utility is a digital tool or calculator designed to visualize mathematical functions and sequences.
It can significantly aid in understanding and interpreting sequences by providing a visual representation of numerical values. To graph a sequence like the one described, a graphing utility helps to plot each term against its corresponding position number, allowing students to see patterns and trends more clearly.
  • Typically, users input a formula or function.
  • The tool creates a graph based on specified values or parameters.
  • This helps students to quickly identify the shape and nature of the sequence or function.
Using a graphing utility saves time and easily illustrates concepts that might be more challenging to grasp from just a formula or tabulated data.
Sequence Terms
In any sequence, sequence terms refer to the individual elements or numbers that make up the sequence. Here, each term, represented by \( a_{n} \), is calculated using a specific formula.
The formula helps find the value of each term based on its position in the sequence.
  • For this exercise, the formula \( a_{n} = \frac{2}{3} n \) is used.
  • To find each term, you substitute the term's position number for \( n \).
  • This substitution computes the value of each specific term.
Understanding how to find individual sequence terms is crucial, as it makes it easier to graph and analyze the behavior of the entire sequence.
Discrete Graph
A discrete graph is a type of graph used to represent data points where numerical values are distinct, typically depicting scenarios where terms or events are countable. Unlike continuous graphs, which illustrate a seamless connection between points, discrete graphs show individual points, each isolated and separate.
  • Each term of the sequence is represented as a single point on the graph.
  • For instance, plotting points (1, \( a_{1} \)), (2, \( a_{2} \)), etc., where each \( a_{n} \) follows the formula \( a_{n} = \frac{2}{3} n \).
  • This format is ideal for sequences and series, where terms are only defined at particular values of \( n \).
Thus, discrete graphs effectively show how each term varies as the position in the sequence changes, helping to visualize the sequence's characteristics.
Term Position
Term position refers to the position number \( n \) in a sequence, which determines each term's value according to the sequence formula. The position is an integral part of understanding sequences, as it's a key factor in calculating the value of the sequence term.
In this arithmetic sequence, the position \( n \) dictates all calculations, illustrating how the sequence progresses.
  • The term position is part of the sequence formula \( a_{n} = \frac{2}{3} n \).
  • By substituting different \( n \) values (i.e., \( n = 1, 2, 3, \ldots \)), each term's value is determined.
  • This pattern of substitution and calculation is fundamental to sequence analysis.
Understanding term positions helps students map sequence progression and study how terms relate to their respective positions.

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Most popular questions from this chapter

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