Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=15-\frac{3}{2} n$$

Step-by-Step Solution

Answer

To graph the first 10 terms of the sequence \(a_{n}=15-\frac{3}{2} n\), you have to generate the term values for \(n = 1, 2, ..., 10\) and then plot these values on a graph with n-value on the x-axis and the corresponding term value on the y-axis. The graph will have separate points, not a continuous line because sequences are made up of distinct terms.

Step 1: Understand the Sequence

The sequence is represented by \(a_{n}=15-\frac{3}{2} n\). This means that to get the value of any term in the sequence, you can substitute the term number (n) into the equation and calculate the corresponding value.

Step 2: Compute the Sequence Values

Start substituting values of n starting from 1 up to 10 into the equation to generate the first 10 terms of the sequence. For example, when \(n = 1\), \(a_{1}= 15-\frac{3}{2} \times 1 = 13.5\). Repeat this step for \(n = 2\),\(n = 3\),...\(n = 10\).

Step 3: Plot the Graph

For each term, represent it as a point on the graph with the term number on the x-axis and the term value on the y-axis. Using a graphing utility, plot all the ten points and observe the pattern that the sequence takes. Remember that sequence graphs have discrete points as each term is distinct from the other.

Key Concepts

Graphing sequences

Graphing sequences is a crucial technique in mathematics that visualizes the growth or decay of sequences. A sequence is simply a list of numbers in a specific order determined by a rule or a formula.
In the case of the sequence given by the formula \(a_{n} = 15 - \frac{3}{2} n\), each term is calculated by substituting \(n\) (the position in the sequence) into the formula. Once you have some terms, you can arrange them in a graph. The x-axis will represent the position \(n\) in the sequence, and the y-axis will display the value \(a_{n}\) of the term.
To graph the sequence:

• Identify the range of \(n\) values, such as 1 through 10 in this case.
• Calculate the sequence values for these \(n\) values.
• Plot the sequence's points where each \(n\) value lines up with its corresponding \(a_{n}\) value.

Discrete Graph FeatureA sequence graph shows distinct points because the values are not continuous; they only exist for specified positions \(n\). This characteristic helps to understand each term's progression clearly, making graphing essential for visual learners.

Sequence terms calculation

Calculating the terms of a sequence involves using its defining formula. Here, you have \(a_{n} = 15 - \frac{3}{2} n\) for calculating sequence terms. To find the value of the first 10 terms:

• Start with \(n = 1\): Substitute 1 into the formula to compute \(a_{1}\). For example, \(a_{1} = 15 - \frac{3}{2} \times 1 = 13.5\).
• Continue substituting values up to \(n = 10\): Repeat the process for each \(n\) value until you’ve calculated through \(a_{10}\).

This repetitive process is simple in terms of the execution, yet it also allows exploration of the pattern or trend in the terms.
By computing these values, you gain not only the list of sequence terms but also insights that assist in predicting subsequent values or understanding the sequence's behavior over a specified range.

Discrete mathematics

Discrete mathematics involves the study of mathematical structures that are fundamentally countable or distinct. Sequences, like many elements in discrete mathematics, involve only integers and often deal with specific, separate numbers rather than a continuous spectrum.
In the context of sequences:

• Each term is distinct per the sequence's formula, with no halfway terms.
• Every term results from a particular position or index, such as the \(n\) in our sequence \(a_{n} = 15 - \frac{3}{2} n\).

Importance of Discrete NatureThe discrete nature of sequences facilitates problems in computer science, game theory, logic, etc., where distinct solutions are necessary. Learning to analyze sequences through formulas like \(a_{n}\) helps harness the discrete structure's power to solve complex problems.

Graphing utilities in mathematics

Graphing utilities are technological tools that help visualize mathematical concepts effortlessly. In sequences' context, graphing utilities take calculated sequence values and plot them rapidly on a graph without manual plotting.
Here are some features and advantages of using graphing utilities:

• Simplification: Quickly input formulas to generate graphs, saving time and reducing error.
• Visualization: See patterns and trends in sequence behaviors more clearly than static tables.
• Analysis: Tools often provide additional features like finding limits or trends in data over intervals.

Graphing utilities remove complexity from calculation-intensive tasks, allowing students to focus on understanding rather than manual plotting.