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Chapter 9: Problem 23

Use a graphing utility to graph the first 10 terms of the sequence. $$ a_{n}=16(-0.5)^{n-1} $$

Short Answer

Expert verified
The graph will start high at 16 (at \( n = 1 \)) and oscillate between positive and negative values, decreasing in absolute value with each term.

Step by step solution

01

Understand the sequence formula

The sequence is given by \( a_{n}=16(-0.5)^{n-1} \). Here, \( n \) is the term number and \( a_{n} \) is the value of the nth term of the sequence.
02

Compute the first 10 terms

To compute the first 10 terms of the sequence, replace \( n \) in the sequence formula with 1, 2, 3,..., 10. This will give the corresponding term value for each term number.
03

Plot the terms

Draw an XY-plane with 'term number' on the x-axis and 'term value' in the y-axis. Put the numbers 1 through 10 on the x-axis and put the corresponding term values computed in step 2 on the y-axis. Plot a point for each term number and its corresponding term value. Connect the points to indicate the progression of the term values.

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Key Concepts

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

Graphing Sequences
When we talk about graphing sequences in mathematics, it involves visualizing the terms of a sequence on a coordinate plane. A sequence is a list of numbers arranged in a specific order where each term is defined by a mathematical formula. For the given sequence, the formula is \( a_{n}=16(-0.5)^{n-1} \). The graph allows us to see how the terms behave or change as the term number (\(n\)) increases.

To graph a sequence:
  • Identify the formula for the sequence. In our case, it's \( a_{n}=16(-0.5)^{n-1} \).
  • Determine the range of term numbers you are interested in. Here, the first 10 terms (\(n=1\) to \(n=10\)) are considered.
  • Calculate the term values using the sequence formula for each term number.
  • On a graph, place term numbers (1, 2, 3, ...) on the horizontal axis (x-axis) and the corresponding term values on the vertical axis (y-axis).
  • Plot each point on the graph and connect them to see the pattern or trend.
This method gives a visual representation of how a sequence behaves, providing insights into patterns such as growth or decay.
Exponential Decay
Exponential decay is a specific type of mathematical function where each term decreases by a consistent rate. Here, the sequence \( a_{n}=16(-0.5)^{n-1} \) involves exponential decay due to the multiplication factor of \(-0.5\). The negative sign along with the base \(0.5\) explains the decay pattern.

Key aspects of exponential decay:
  • The base (here \(-0.5\)) is between \(-1\) and \(0\), which causes the sequence to alternate in sign, giving it an oscillating behavior along with decay.
  • As \(n\) increases, the term values approach zero because the absolute value of the base is less than 1.
  • The initial amount, here \(16\), represents the starting point before decay begins. Each subsequent term decreases in value relative to the last.
Recognizing exponential decay can help you predict long-term behavior in sequences, ensuring clear expectations for future terms.
Plotting Points on a Graph
Plotting points on a graph is a fundamental skill in mathematics to interpret and analyze sequences effectively. Each point on the graph corresponds to a term number and its respective term value based on the sequence formula.

To plot points on a graph:
  • First, draw your coordinate plane, ensuring you have a clear x-axis for term numbers and a y-axis for term values.
  • Determine your scale based on the range of term numbers and values you have. This ensures each point will be placed accurately.
  • Plot each term by locating its term number on the x-axis and its corresponding value on the y-axis.
  • After plotting all points, connect them in sequential order to show the progression or trend of the sequence.
This visual tool is crucial for understanding sequences, as it provides a direct way to spot any repeating patterns, increasing or decreasing trends, and other notable behaviors.

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Most popular questions from this chapter

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Use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series. $$ f(x)=2\left[\frac{1-(0.8)^{x}}{1-0.8}\right] \quad \sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n} $$
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