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Chapter 9: Problem 72

Graphing the Terms of a Sequence In Exercises \(69 - 72 ,\) use a graphing utility to graph the first 10 terms of the sequence. (Assume that \(n\) begins with \(1 . )\) $$ a _ { n } = - 0.3 n + 8 $$

Short Answer

Expert verified
To graph the terms of the sequence, calculate the values of the first ten terms using \(a _ { n } = - 0.3 n + 8\), make a list of ordered pairs \((n, a_n)\), plot the points in the pairs, and draw lines between consecutive points to help visualize the sequence trend.

Step by step solution

01

Calculate sequence values

As \(n\) ranges from 1 to 10, calculate the corresponding values of \(a_n\) using the given formula \(a _ { n } = - 0.3 n + 8\). This step will provide the terms of the sequence.
02

Create a list of ordered pairs

Since a sequence is a function on the set of positive integers, each term can be represented as an ordered pair of its index and value. Create a list of ordered pairs \((n, a_n)\), for \(n\) ranging from 1 to 10. The list will look something like: \((1, a_1), (2, a_2), (3, a_3) , ..., (10, a_{10})\). This list will serve as a guide to plot the points.
03

Plot the points

Create a graph with \(n\) on the x-axis and \(a_n\) on the y-axis. Plot each of the calculated points from the created list of ordered pairs on the graph. Use the order pairs list as a guide where each pair is a coordinate on the graph. The first value of each pair is the x-coordinate, and the second value is the y-coordinate.
04

Connect the points

Once all points have been plotted on the graph, connect them with line segments to better visualize the trend of the sequence. Although in a mathematical sequence there is no line, to help visualize the sequence trend, we include the line connecting the points in the graph.

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Key Concepts

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

Understanding Ordered Pairs
An ordered pair is a fundamental concept in mathematics used to describe a point in two-dimensional space. Each ordered pair is typically written in the form
  • \((x, y)\)
where \(x\) represents the horizontal coordinate (or the index in a sequence), and \(y\) is the vertical coordinate (the sequence value). In the context of sequences, ordered pairs help us to map each term of the sequence to its position. This means that for each index \(n\), you calculate the sequence value \(a_n\), and you can write it out as an ordered pair
  • \((n, a_n)\)
For example, if your sequence starts with 1 and the first term is 7.7, the representation would be the ordered pair \((1, \, 7.7)\). Ordered pairs can then be used to easily plot points on a graph, helping you visualize the sequence's behavior.
Using a Graphing Utility
A graphing utility is an amazing tool that makes plotting sequences and other mathematical functions much more straightforward. These can be software, online applications, or physical calculators. With a graphing utility, you'll generally follow these steps:
  • Input the sequence formula or ordered pairs.
  • Choose the range for \(n\), like 1 to 10 in this case.
  • Observe the plotted graph on the screen.
Using a graphing utility saves time and offers clearer visualization, especially helpful for complex sequences. Once you input your ordered pairs, the utility can plot them instantly. Some utilities also provide options to enhance your graph with features like grid lines, labels, or different point shapes and sizes. This allows for a more comprehensive analysis and understanding of your data.
Connecting Points to Visualize Trends
When you plot the ordered pairs of a sequence on a graph, you'll see what appears to be scattered dots or points. These dots represent each term in your sequence, aligned along the axes. To better illustrate how the sequence behaves as \(n\) changes, it's helpful to connect these points with line segments. While a sequence is typically discrete, connecting the points provides a clearer picture of the pattern or trend they form. For instance:
  • It could help identify if your sequence is increasing, decreasing, or following any specific pattern.
  • It enhances visual comprehension by showing slope or rate of change between terms.
Remember, connecting points is merely a visual aid, not a transformation of the sequence into a continuous function. It's a helpful tool for spotting trends within the data and communicating them effectively.

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Most popular questions from this chapter

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