Problem 21 - Algebra and Trigonometry

Question

\(19-24\) . 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. Graph these values as points.

Step 1: Identify the Sequence Formula

The formula given for the sequence is \(a_n = \frac{12}{n}\). This means that for each term \(a_n\), we substitute \(n\) with its corresponding term number.

Step 2: Calculate the First 10 Terms

To find the first 10 terms, we substitute \(n\) from 1 to 10 into the formula \(a_n = \frac{12}{n}\). - When \(n = 1\), \(a_1 = \frac{12}{1} = 12\). - When \(n = 2\), \(a_2 = \frac{12}{2} = 6\). - When \(n = 3\), \(a_3 = \frac{12}{3} = 4\). - When \(n = 4\), \(a_4 = \frac{12}{4} = 3\). - When \(n = 5\), \(a_5 = \frac{12}{5} = 2.4\). - When \(n = 6\), \(a_6 = \frac{12}{6} = 2\). - When \(n = 7\), \(a_7 = \frac{12}{7} \approx 1.71\). - When \(n = 8\), \(a_8 = \frac{12}{8} = 1.5\). - When \(n = 9\), \(a_9 = \frac{12}{9} \approx 1.33\). - When \(n = 10\), \(a_{10} = \frac{12}{10} = 1.2\).

Step 3: Graph the Sequence Terms

To graph the sequence, plot each term on a graph using \(n\) as the x-coordinate and \(a_n\) as the y-coordinate. The coordinates are: - (1, 12) - (2, 6) - (3, 4) - (4, 3) - (5, 2.4) - (6, 2) - (7, 1.71) - (8, 1.5) - (9, 1.33) - (10, 1.2) This will show how the value of \(a_n\) decreases as \(n\) increases.

Key Concepts

Sequence Formula

A sequence can be thought of as a list of numbers in a specific order, often defined by a formula. The given formula for this sequence is \(a_n = \frac{12}{n}\). In this formula, \(a_n\) represents the nth term in the sequence, and \(n\) is the position of the term in the sequence.
This means that to find any term in the sequence, you just replace \(n\) with the specific term number you're interested in.

For example, to find the first term \(a_1\), simply substitute \(n = 1\) into the formula, so:\[ a_1 = \frac{12}{1} = 12 \].

• The formula is a simple way to describe each term mathematically.
• Understanding the sequence formula enables you to predict future terms without listing all previous ones.

The idea here is that this formula can be repeated, providing a tractable system for all terms.

Term Calculation

In sequences, each individual number is known as a term. To find terms of a sequence, we use the sequence formula. Using the formula \(a_n = \frac{12}{n}\), we calculated the first 10 terms. Each term is found by substituting the value of \(n\) ranging from 1 to 10.

Calculations are as follows:

• For \(n = 1\), \(a_1 = 12\)
• For \(n = 2\), \(a_2 = 6\)
• For \(n = 3\), \(a_3 = 4\)
• For \(n = 4\), \(a_4 = 3\)
• For \(n = 5\), \(a_5 = 2.4\)
• For \(n = 6\), \(a_6 = 2\)
• For \(n = 7\), \(a_7 \approx 1.71\)
• For \(n = 8\), \(a_8 = 1.5\)
• For \(n = 9\), \(a_9 \approx 1.33\)
• For \(n = 10\), \(a_{10} = 1.2\)

It's clear that as \(n\) becomes larger, the term size decreases. Calculating each term is crucial for understanding how the sequence behaves over time. These calculations highlight the relations between the position of terms and their values.

Graphing Sequences

Graphing sequences illustrates the behavior of terms visually, offering insights that numbers alone might not reveal. To graph the sequence from the formula \(a_n = \frac{12}{n}\), each term calculated becomes a point on the graph. The x-coordinate of each point corresponds to the term number \(n\), and the y-coordinate is the term \(a_n\).

The first 10 points are as follows:

• (1, 12)
• (2, 6)
• (3, 4)
• (4, 3)
• (5, 2.4)
• (6, 2)
• (7, 1.71)
• (8, 1.5)
• (9, 1.33)
• (10, 1.2)

Placing these points on a graph shows a pattern where \(a_n\) decreases as \(n\) increases. Graphs are helpful tools for visual learners and assist in predicting the behavior of the sequence as \(n\) continues to grow. By observing the plotted graph, students can grasp the relationship between term placement and term size more effectively.