Problem 23 - Precalculus: Mathematics for Calculus

Question

Use a graphing calculator to do the following. (a) Find the first ten terms of the sequence. (b) Graph the first ten terms of the sequence. \(a_{n}=\frac{12}{n}\)

Step-by-Step Solution

Answer

Terms: 12, 6, 4, 3, 2.4, 2, 1.71, 1.5, 1.33, 1.2; Graph shows decreasing values.

Step 1: Determine Sequence Formula

Understand the formula given in the exercise: \(a_n = \frac{12}{n}\). This formula will be used to calculate the first ten terms of the sequence.

Step 2: Substitute Values for n

To find the first ten terms, substitute values 1 through 10 into the formula \(a_n = \frac{12}{n}\). Calculate each term individually.

Step 3: Calculate First Ten Terms

Calculate:- For \(n = 1\), \(a_1 = \frac{12}{1} = 12\)- For \(n = 2\), \(a_2 = \frac{12}{2} = 6\)- For \(n = 3\), \(a_3 = \frac{12}{3} = 4\)- For \(n = 4\), \(a_4 = \frac{12}{4} = 3\)- For \(n = 5\), \(a_5 = \frac{12}{5} = 2.4\)- For \(n = 6\), \(a_6 = \frac{12}{6} = 2\)- For \(n = 7\), \(a_7 = \frac{12}{7} \approx 1.71\)- For \(n = 8\), \(a_8 = \frac{12}{8} = 1.5\)- For \(n = 9\), \(a_9 = \frac{12}{9} \approx 1.33\)- For \(n = 10\), \(a_{10} = \frac{12}{10} = 1.2\)The first ten terms are: 12, 6, 4, 3, 2.4, 2, 1.71, 1.5, 1.33, and 1.2.

Step 4: Graph the Terms of the Sequence

Use a graphing calculator to plot these terms on a graph. The x-axis represents the term number (n from 1 to 10), and the y-axis represents the value of the terms (\(a_n\)). Plot the points (1, 12), (2, 6), (3, 4), (4, 3), (5, 2.4), (6, 2), (7, 1.71), (8, 1.5), (9, 1.33), and (10, 1.2). Observe how the graph decreases as \(n\) increases.

Key Concepts

Graphing Calculator

A graphing calculator is a handy tool that can help visualize mathematical concepts. Unlike a regular calculator, a graphing calculator can display graphs of equations and sequences, which aids significantly in understanding their behavior.

Using a graphing calculator involves:

• Entering the formula of a sequence or equation
• Setting the range for your variables (e.g., the first ten terms of a sequence)
• Viewing the plotted graph to analyze data visually

For our sequence, plugging in the sequence formula into the calculator allows us to see how each term corresponds to different values of \(n\). This gives a visual representation of how the sequence decreases as the term number increases.When using a graphing calculator, ensure the settings are correct—for instance, setting the axes' limits according to the values being calculated. This ensures the graph displays properly and gives an accurate picture of the data. These details can greatly enhance comprehension when studying sequences or other mathematical concepts.

Sequence Formula

Understanding a sequence formula is a critical step in working with sequences in mathematics. In our example, the formula is given as \(a_n = \frac{12}{n}\). This formula helps determine the terms of the sequence by specifying the rule for their calculation.

The key parts of a sequence formula include:

• \(a_n\) - the term of the sequence for a particular \(n\)
• The mathematical expression on the right-hand side, which determines how the terms are calculated

The formula \(a_n = \frac{12}{n}\) tells us that each term is calculated by dividing 12 by \(n\), where \(n\) is the position number of the term in the sequence. Applying this formula with different values of \(n\) provides a list of terms that comprise the sequence, showing how each term changes depending on \(n\). Recognizing these elements helps in understanding how different sequences form unique patterns, which can reveal interesting properties and behaviors when graphed or analyzed.

Graphing Sequences

Graphing sequences is an effective way to gain insight into their behavior over a range of terms. When we graph the sequence described by \(a_n = \frac{12}{n}\), each point on the graph represents a term in the sequence, with \(n\) as the x-coordinate and \(a_n\) as the y-coordinate.

When graphing sequences, consider the following steps:

• Identify the terms to graph (e.g., the first ten terms)
• Set up your graph with the x-axis as term number and the y-axis as the term value
• Plot each point and observe the pattern formed

The plotted graph for \(a_n = \frac{12}{n}\) will show a downward trend, indicating that as \(n\) increases, the terms \(a_n\) decrease. This visual can help underline the general characteristics of the sequence such as convergence or divergence.

This visual representation is crucial for understanding more abstract mathematical concepts as it translates numbers into observable patterns, making sequence analysis intuitive and accessible.