Problem 31 - 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. $$ 7.5,6,4.5,3,1.5 $$

Step-by-Step Solution

Answer

8 terms: 7.5, 6, 4.5, 3, 1.5, 0, -1.5, -3; graph straight line; formula: \(a_n = 7.5 + (n-1)(-1.5)\).

Step 1: Identifying the Sequence

The sequence is an arithmetic sequence, which means that each term is obtained by adding a constant difference to the previous term. Here, the first five terms are given: 7.5, 6, 4.5, 3, 1.5.

Step 2: Finding the Common Difference

To find the common difference, subtract the second term from the first term: \(6 - 7.5 = -1.5\). This common difference of \(-1.5\) applies to each consecutive pair of terms.

Step 3: Extending the Sequence

Using the common difference \(-1.5\), calculate the next three terms after the 1.5: - Next term: \(1.5 - 1.5 = 0\) - Following term: \(0 - 1.5 = -1.5\) - Following term: \(-1.5 - 1.5 = -3\) This gives us the first eight terms: 7.5, 6, 4.5, 3, 1.5, 0, -1.5, -3.

Step 4: Numerical Representation

The numerical representation of the sequence is a listing of numbers: 7.5, 6, 4.5, 3, 1.5, 0, -1.5, -3.

Step 5: Graphical Representation

Create a graph with the terms of the sequence as points. The x-axis will represent the term number (from 1 to 8), and the y-axis will represent the term values. Plot the points (1, 7.5), (2, 6), (3, 4.5), (4, 3), (5, 1.5), (6, 0), (7, -1.5), (8, -3) and connect them with a straight line because it's an arithmetic sequence.

Step 6: Symbolic Representation

For an arithmetic sequence, we use the formula for the nth term: \( a_n = a_1 + (n-1) imes d \), where \( a_1 = 7.5 \) is the first term and \( d = -1.5 \) is the common difference. The formula becomes \( a_n = 7.5 + (n-1)(-1.5) \).

Key Concepts

Common Difference

In an arithmetic sequence, every term after the first is obtained by adding a fixed number to the previous term. This fixed number is known as the "common difference." The common difference is a vital element in an arithmetic sequence, as it governs how the sequence progresses. For the sequence 7.5, 6, 4.5, 3, 1.5 given in the exercise, finding this common difference is our starting point. By subtracting the second term from the first, we identify that the common difference is \(-1.5\). This means each number in the sequence is \(-1.5\) less than the number before it.

• To find the common difference, just subtract any term from the term that follows it.
• The common difference for this sequence is constant across all term pairs.

Recognizing the common difference is crucial when extending the sequence further or analyzing its properties.

Graphical Representation

Visualizing an arithmetic sequence can provide insight into its behavior. The graphical representation involves plotting the sequence terms on a graph. For our sequence, we plot the terms on a coordinate plane.

• The x-axis represents the term numbers, like 1, 2, 3, up to 8 in our extended sequence.
• The y-axis shows the values of these terms: 7.5, 6, 4.5, etc.

When you plot these points: (1, 7.5), (2, 6), (3, 4.5), (4, 3), (5, 1.5), (6, 0), (7, -1.5), and (8, -3), you can connect them in a straight line. This straight line is characteristic of arithmetic sequences.
Observing a straight-line graph confirms the constant decrement defined by the common difference. It also offers a clear view of how each term fits relative to others in terms of both value and position.

Symbolic Representation

Understanding arithmetic sequences isn't complete without expressing them through formulas. The symbolic representation uses a general formula for the nth term: \(a_n = a_1 + (n-1) \times d\). This formula allows us to calculate any term's value based on its position in the sequence. Here's how it translates for the sequence from the exercise:

• \(a_1 = 7.5\), which is the first term.
• \(d = -1.5\), the common difference.

Substituting these values into the formula gives \(a_n = 7.5 + (n-1)(-1.5)\). This equation can be used to find any term in the sequence without needing to list all the terms from the beginning.
This symbolic form is powerful because it abstracts the arithmetic sequence into a manageable mathematical expression, useful for analysis and computation, enhancing both understanding and efficiency when dealing with these sequences.