Problem 82
Question
In Exercises 79 - 82, use a graphing utility to graph the first \( 10 \) terms of the sequence. (Assume \( n \) that begins with \( 1 \).) \( a_n = -0.3n + 8 \)
Step-by-Step Solution
Verified Answer
The first ten terms of the sequence are obtained by using \( a_n = -0.3n + 8 \). When graphed, these terms will form a descending linear pattern, reflecting the negative coefficient of the sequence.
1Step 1: Generate the First 10 Terms
The first task is to generate the first \( 10 \) terms of the sequence. This can be done by plugging \( n \) values from \( 1 \) to \( 10 \) into the sequence definition \( a_n = -0.3n + 8 \).
2Step 2: Plot the Points on a Grid
With these computed values for \( a_n \), plot the points on a Cartesian grid. Use the term number \( n \) as the x-coordinate and the term value \( a_n \) as the y-coordinate for each point.
3Step 3: Visualize the Sequence
The last step is to connect these points with line segments. This will visualize how the sequence progresses as \( n \) increases. The trend of the sequence should be identifiable from the graph.
Key Concepts
Arithmetic SequencesGraphing Utility UseSequence Convergence
Arithmetic Sequences
An arithmetic sequence is a list of numbers with a common difference between successive terms. In other words, each term after the first is obtained by adding a fixed amount, known as the common difference, to the previous term. For example, the sequence 2, 5, 8, 11, ... is arithmetic because each term is 3 more than the previous term.
In the given exercise, \( a_n = -0.3n + 8 \) describes an arithmetic sequence because the terms of the sequence decrease by a constant value of 0.3 each time the \( n \) increases by one. The sequence begins with \( a_1 = 7.7 \) and \( a_2 = 7.4 \) indicating that the common difference is \( -0.3 \)\.
Understanding arithmetic sequences is fundamental in various mathematical concepts and real-life applications, such as calculating savings over time, determining patterns in nature, or even predicting trends in social phenomena.
In the given exercise, \( a_n = -0.3n + 8 \) describes an arithmetic sequence because the terms of the sequence decrease by a constant value of 0.3 each time the \( n \) increases by one. The sequence begins with \( a_1 = 7.7 \) and \( a_2 = 7.4 \) indicating that the common difference is \( -0.3 \)\.
Understanding arithmetic sequences is fundamental in various mathematical concepts and real-life applications, such as calculating savings over time, determining patterns in nature, or even predicting trends in social phenomena.
Graphing Utility Use
Graphing utilities, such as calculators or software, allow for the visualization of mathematical concepts, including sequences. These tools can be highly beneficial when you're working with sequences because they provide a visual representation that can make it easier to understand and analyze the behavior of a sequence.
When instructed to use a graphing utility in an exercise, the goal is typically to plot each term of the sequence against its term number to see the trend. For the sequence in the exercise, \( a_n = -0.3n + 8 \)\, you would input the first ten values of \( n \) and observe the resulting points on the graph.
By using a graphing utility, learners can quickly see if the sequence is increasing or decreasing, and by how much. With the exercise in question, the graphing utility would show a linear decrease, reinforcing the concept of an arithmetic sequence with a negative common difference.
When instructed to use a graphing utility in an exercise, the goal is typically to plot each term of the sequence against its term number to see the trend. For the sequence in the exercise, \( a_n = -0.3n + 8 \)\, you would input the first ten values of \( n \) and observe the resulting points on the graph.
By using a graphing utility, learners can quickly see if the sequence is increasing or decreasing, and by how much. With the exercise in question, the graphing utility would show a linear decrease, reinforcing the concept of an arithmetic sequence with a negative common difference.
Sequence Convergence
In mathematics, sequence convergence refers to the behavior of a sequence as the term number increases indefinitely. If the sequence approaches a specific value as \( n \) gets larger, the sequence is said to be convergent. Conversely, if the terms of the sequence continue to increase or decrease without approaching a single value, the sequence is divergent.
The arithmetic sequence given in the exercise, \( a_n = -0.3n + 8 \)\, is an example of a divergent sequence. As \( n \) grows, the value of \( a_n \) keeps decreasing by 0.3 and will continue to do so without settling towards any particular number, thus highlighting the concept of divergence in the given sequence.
Understanding whether a sequence converges or diverges is crucial for further studies in calculus, such as series and integrals, and has implications in fields ranging from physics to finance.
The arithmetic sequence given in the exercise, \( a_n = -0.3n + 8 \)\, is an example of a divergent sequence. As \( n \) grows, the value of \( a_n \) keeps decreasing by 0.3 and will continue to do so without settling towards any particular number, thus highlighting the concept of divergence in the given sequence.
Understanding whether a sequence converges or diverges is crucial for further studies in calculus, such as series and integrals, and has implications in fields ranging from physics to finance.
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Problem 82
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