Problem 21 - Precalculus Mathematics for Calculus

Question

Use a graphing calculator to do the following. (a) Find the first 10 terms of the sequence. (b) Graph the first 10 terms of the sequence. $$a_{n}=\frac{12}{n}$$

Step-by-Step Solution

Answer

The first 10 terms are: 12, 6, 4, 3, 2.4, 2, 1.71, 1.5, 1.33, 1.2. These terms can be plotted using a graphing calculator.

Step 1: Understand the Sequence Rule

The sequence is defined by the rule \( a_n = \frac{12}{n} \). This means that the value of the sequence at each term \( n \) is given by dividing 12 by \( n \).

Step 2: Calculate the First 10 Terms

To find the first 10 terms, substitute \( n = 1, 2, \, \ldots, \, 10 \) into the formula:- 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 \( n = 10 \), \( a_{10} = \frac{12}{10} = 1.2 \).

Step 3: Prepare the Graph

Create a coordinate plane with the term number \( n \) on the horizontal axis and the sequence value \( a_n \) on the vertical axis. Plot each of the first 10 terms as calculated earlier.

Step 4: Graph the Terms Using Calculator

Using a graphing calculator, input each pair of coordinates \((n, a_n)\) as a point on the graph. This can typically be done in the 'scatter plot' mode of the calculator. Ensure the points are clear and accurately represent the sequence.

Key Concepts

Sequence

In mathematics, a sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a "term." Understanding the defining rule is crucial because it tells us how to get from one term to the next. In the case of the sequence defined by the rule \( a_n = \frac{12}{n} \), the pattern follows a specific calculation: divide the constant 12 by each term number \( n \).

• The first term \( a_1 \) is 12 because 12 divided by 1 is 12.
• As \( n \) increases, the term \( a_n \) decreases because 12 is being divided by a larger number.

Sequences are common in various fields like mathematics, science, and finance, and understanding them helps predict future behaviors or trends.

Terms of a Sequence

The terms of a sequence refer to the individual elements, or numbers, that make up the sequence. Each term is identified by its position in the sequence, indicated by \( n \). In our example, the terms were calculated using the formula \( a_n = \frac{12}{n} \). Let's have a closer look at some characteristics of terms:

• Each term is distinct and determined by substituting \( n \) into the sequence's formula.
• The first 10 terms are: 12, 6, 4, 3, 2.4, 2, 1.71, 1.5, 1.33, and 1.2. Notice how they gradually decrease.

Ultilizing these terms can reveal patterns in the data, which can be helpful in analysis and prediction.
Your understanding of terms makes you capable of not just solving the sequence, but also applying it in real-world contexts.

Scatter Plot

A scatter plot is a powerful graphical representation used to display data points on a coordinate plane. It is especially effective for showing how one variable is affected by the other, often revealing patterns or trends in data. When dealing with sequences, a scatter plot can illuminate the relationship between the sequence number \( n \) and the corresponding term \( a_n \).
To create a scatter plot:

• Use the graphing calculator’s 'scatter plot' mode, which allows you to input the data points.
• Each point on the scatter plot consists of two coordinates: the position \( n \) on the x-axis and the term \( a_n \) on the y-axis.
• For our sequence, you plot points like (1,12), (2,6), and so forth, up to the tenth term.

Scatter plots help in visualizing how the values of a sequence converge or diverge, making it easier to interpret the behavior of the sequence visually. This visual aid is invaluable, particularly when analyzing complex patterns not immediately obvious by observation alone.