Problem 33 - 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

Numerical: 5.1, 5.5, 5.9, 6.3, 6.7, 7.1, 7.5, 7.9; Graphical: plot as linear graph; Symbolic: \(a_n = 0.4n + 4.7\).

Step 1: Recognize the Arithmetic Sequence

From the given terms 5.1, 5.5, 5.9, 6.3, and 6.7, identify the common difference. Subtract each term from the next: 5.5 - 5.1 = 0.4, 5.9 - 5.5 = 0.4, etc. The common difference is 0.4.

Step 2: Extend the Sequence Numerically

Using the common difference of 0.4, continue the sequence by adding 0.4 to the last given term, 6.7. This gives: 7.1, 7.5, and 7.9, making a total of eight terms: 5.1, 5.5, 5.9, 6.3, 6.7, 7.1, 7.5, 7.9.

Step 3: Create a Graph of the Sequence

Plot the sequence terms on a graph with the x-axis representing the term number (1 to 8) and the y-axis the term values. Start at (1, 5.1) and end at (8, 7.9), plotting each subsequent term. The graph should show a straight line with a positive slope, confirming the arithmetic nature of the sequence.

Step 4: Establish the Symbolic Equation

For an arithmetic sequence, the nth term is given by the formula \(a_n = a_1 + (n-1) imes d\), where \(a_1 = 5.1\) and \(d = 0.4\). Inputting these into the formula gives \(a_n = 5.1 + (n-1) imes 0.4\). Simplifying, \(a_n = 0.4n + 4.7\).

Step 5: Validate with Additional Terms

Use the symbolic equation to find additional terms. For example, for the 6th term \(a_6 = 0.4(6) + 4.7 = 7.1\), confirming the sequence consistency and correctness.

Key Concepts

Numerical Representation

Numerical representation in an arithmetic sequence means listing out the terms in order. An arithmetic sequence changes constantly by the same amount, known as the common difference. Here, the sequence starts with 5.1 and each subsequent term adds 0.4. The first five terms are given:

• 5.1
• 5.5
• 5.9
• 6.3
• 6.7

To find more terms, you add 0.4 to the last term, 6.7, to find the next term, which is 7.1. Continuing this process, you get:

• 7.1
• 7.5
• 7.9

This makes a complete set of eight terms: 5.1, 5.5, 5.9, 6.3, 6.7, 7.1, 7.5, and 7.9. These terms collectively represent the numerical aspect of the sequence by showing the actual numbers and their uniform interval.

Graphical Representation

To visually understand the arithmetic sequence, you can create a graph. A graph provides a clear visualization of the pattern, showing how each term relates to its position in the sequence. Here's how to do it:

• Label the x-axis with term numbers (1 to 8)
• Label the y-axis with the term values

The points to plot represent each term like so:

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

Connecting these points creates a straight line with a positive slope. This line demonstrates the constant increment between the terms and confirms that the sequence is indeed arithmetic. It's a visually intuitive way to grasp the concept of a sequence's uniform progression.

Symbolic Representation

Symbolic representation captures the pattern of an arithmetic sequence in a mathematical formula. This formula can be used to find any term in the sequence without listing them.The formula for the nth term in an arithmetic sequence is: \[ a_n = a_1 + (n-1) \cdot d \] where:

• \( a_1 = 5.1 \) (the first term)
• \( d = 0.4 \) (the common difference)

Substituting these values into the formula gives: \[ a_n = 5.1 + (n-1) \cdot 0.4 \]Simplifying, we obtain:\[ a_n = 0.4n + 4.7 \]This symbolic form provides a quick way to compute any term in the sequence. For example, to find the 6th term: \[ a_6 = 0.4 \times 6 + 4.7 = 7.1 \]This approach ensures accuracy and is especially helpful for finding terms that aren't at the beginning or when the sequence is large.