Problem 32 - College Algebra with Modeling and Visualization

Question

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$5.1,5.5,5.9,6.3,6.7$$

Step-by-Step Solution

Answer

First term is 5.1, common difference is 0.4; terms: 5.1, 5.5, ..., 7.9; formula: \( a_n = 5.1 + (n-1) \times 0.4 \).

Step 1: Identify the first term and common difference

The first term of the sequence is already given as \( a_1 = 5.1 \). To find the common difference, \( d \), subtract the first term from the second term (i.e., \( 5.5 - 5.1 = 0.4 \)). Thus, the common difference is \( d = 0.4 \).

Step 2: Calculate additional terms

Continue from the last given term, \( a_5 = 6.7 \), to find three more terms. - \( a_6 = a_5 + d = 6.7 + 0.4 = 7.1 \)- \( a_7 = a_6 + d = 7.1 + 0.4 = 7.5 \)- \( a_8 = a_7 + d = 7.5 + 0.4 = 7.9 \)The sequence now includes eight terms: \( 5.1, 5.5, 5.9, 6.3, 6.7, 7.1, 7.5, 7.9 \).

Step 3: Create the numerical representation

List the first eight terms of the sequence: \( 5.1, 5.5, 5.9, 6.3, 6.7, 7.1, 7.5, 7.9 \).

Step 4: Plot the graphical representation

On a graph, plot the eight points corresponding to the terms:1. \((1, 5.1)\)2. \((2, 5.5)\)3. \((3, 5.9)\)4. \((4, 6.3)\)5. \((5, 6.7)\)6. \((6, 7.1)\)7. \((7, 7.5)\)8. \((8, 7.9)\)Connect the points with line segments to show the linearity of the arithmetic sequence.

Step 5: Define the symbolic (general) form

The general term of an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \). Substitute \( a_1 = 5.1 \) and \( d = 0.4 \) into this formula to get:\[ a_n = 5.1 + (n-1) \times 0.4 \]

Key Concepts

Numerical Representation

To understand the numerical representation of an arithmetic sequence, let's dive into how the sequence is constructed step by step. In the case of our example, the sequence begins with the first term, which is given as \( a_1 = 5.1 \). The common difference \( d \) is the increment added to each term to arrive at the next one, calculated by subtracting the first term from the second term: \( 5.5 - 5.1 = 0.4 \). Thus, each term increases by 0.4.
The eight terms of our sequence are:

• \(5.1\)
• \(5.5\)
• \(5.9\)
• \(6.3\)
• \(6.7\)
• \(7.1\)
• \(7.5\)
• \(7.9\)

This representation is simply a list of the calculated terms and shows how each is derived from the previous one by adding the common difference of 0.4. It provides a straightforward view of the sequence and makes it easy to predict subsequent terms.

Graphical Representation

The graphical representation of an arithmetic sequence provides a visual insight that complements the numerical data. By plotting each term of the sequence as a point on a graph, we can clearly observe the linear pattern of an arithmetic sequence.
For our arithmetic sequence, the graph points are:

• \((1, 5.1)\)
• \((2, 5.5)\)
• \((3, 5.9)\)
• \((4, 6.3)\)
• \((5, 6.7)\)
• \((6, 7.1)\)
• \((7, 7.5)\)
• \((8, 7.9)\)

When these points are plotted and connected, they form a straight line, which is characteristic of an arithmetic sequence. Each point corresponds to the term of the sequence, where the x-coordinate represents the term number and the y-coordinate represents the term value. This graphical representation is not only useful for understanding the constant rate of change but also for identifying any potential errors in calculations.

Symbolic Representation

The symbolic representation of an arithmetic sequence provides a formulaic method to determine any term within the sequence. This is essential for extending the sequence without manually calculating each term.
The general formula for an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, and \( d \) is the common difference. Apply this to our sequence by substituting \( a_1 = 5.1 \) and \( d = 0.4 \): \[ a_n = 5.1 + (n-1) \times 0.4 \] This formula allows you to calculate any term in the sequence quickly. For example, to find the 9th term, substitute \( n = 9 \) into the formula: \[ a_9 = 5.1 + (9-1) \times 0.4 = 8.3 \] The symbolic form captures the essence of the sequence succinctly and can be generalized beyond the terms initially computed.