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Chapter 9: Problem 53

Use a graphing utility to graph the first 10 terms of the sequence. $$a_{n}=20(-1.25)^{n-1}$$

Short Answer

Expert verified
The graph of the sequence \(a_{n}=20(-1.25)^{n-1}\) will display an alternating pattern due to the exponent's base being negative. This is reflected in the alternating positive and negative term values.

Step by step solution

01

Understanding the sequence

Consider the given sequence \(a_{n}=20(-1.25)^{n-1}\). Here, \(a_{n}\) is the nth term, n is the term number starting from 1.
02

Calculating the terms

Now, calculate the values for the first ten terms of the sequence using the formula \(a_{n}=20(-1.25)^{n-1}\). This will result in the list of values for the first 10 terms.
03

Plotting the sequence on a graph

Using a graphing utility, each term \(a_{n}\) should be marked on the y-axis and its corresponding ordinal number n, which represents the term position, should be marked on the x-axis. Connect the points to visualize the progression of the sequence.

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Key Concepts

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

Graphing Utilities
Graphing utilities are tools that help us visualize mathematical concepts by creating graphs. These can range from simple online graphing calculators to more complex software that provides a variety of options for graph representation.
Let's explore how these utilities function and their importance:
  • **Ease of Plotting: ** With graphing utilities, you can easily input complex functions and sequences. This allows you to focus on analyzing the graph rather than the mechanics of drawing it
  • **Immediate Feedback: ** By plotting graphs, you get immediate visual feedback, which is crucial for understanding how sequences like exponential ones behave over time
  • **Accuracy: ** These tools provide precise plots which help in presenting data without human error that may occur during manual graph drawing
  • **Interactivity: ** Most graphing utilities offer interactive features like zooming and dynamic adjustments, enabling a deeper exploration of mathematical relationships
In the context of sequences, using a graphing utility allows us to see how the sequence evolves as we progress through terms. This can be particularly enlightening for sequences such as the exponential one in our exercise.
Exponential Sequences
Exponential sequences are sequences where each term is an exponential function of the term number. These sequences are characterized by rapid growth or decay depending on whether the base of the exponent is greater or less than one, respectively.
In the sequence given by the formula \(a_{n}=20(-1.25)^{n-1}\), there are a few interesting features to note:
  • **Constant Ratio: ** Exponential sequences have a constant ratio between consecutive terms. Here, each term is multiplied by -1.25 to get the next term
  • **Sign Alteration: ** Since our base is negative, the sequence alternates signs, meaning the terms will be positive and negative in an alternating fashion
  • **Growth/Decay: ** With a base of -1.25, the absolute size of terms will grow owing to 1.25 being greater than 1, but the negative nature gives them a zigzag pattern of ups and downs
Understanding the behavior of exponential sequences equips learners with the skills to analyze data that involve exponential changes, like population growth or radioactive decay, which similarly exhibit rapid scaling properties.
Plotting Graphs
Plotting graphs involves showing data or mathematical functions visually. For sequences, it means placing each term on a grid to see the progression. The exercise asks us to plot the first ten terms of the sequence \(a_{n}=20(-1.25)^{n-1}\). Here's how you can interpret and do that:
  • **Determine Points: ** First, calculate the ten terms using the given formula, then label each term's value as a coordinate \((n, a_{n})\), where \(n\) is the term number
  • **Setting Axes: ** The x-axis represents the term number \(n\), while the y-axis shows the sequence value \(a_{n}\)
  • **Plot Points: ** On the graph, plot each of the points. You may notice the alternating up and down nature due to the negative base
  • **Connect Dots: ** While it's helpful to connect the dots to see the pattern visually, remember the sequence is technically a set of discrete values
Plotting helps you see trends and patterns that are not immediately obvious through the numbers alone. Through careful plotting, the nuances of sequence behavior come to life, such as the exponential growth and alternation in our example. This visual representation can greatly aid in your understanding of sequences.

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Most popular questions from this chapter

A fire company keeps two rescue vehicles. Because of the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available when needed is \(90 \% .\) The availability of one vehicle is independent of the availability of the other. Find the probability that (a) both vehicles are available at a given time, (b) neither vehicle is available at a given time, and (c) at least one vehicle is available at a given time.

Determine whether the statement is true or false. Justify your answer.The Binomial Theorem could be used to produce each row of Pascal's Triangle.

Can your graphing utility evaluate \(_{100} P_{80} ?\) If not, then explain why.

Assume that the probability of the birth of a child of a particular sex is \(50 \% .\) In a family with four children, what is the probability that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy?

Prove the property for all integers \(r\) and \(n\) where \(0 \leq r \leq n\).The sum of the numbers in the \(n\) th row of Pascal's Triangle is \(2^{n}\).
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