Problem 36 - College Algebra with Modeling and Visualization

Question

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$32,-8,2,-\frac{1}{2}, \frac{1}{8}$$

Step-by-Step Solution

Answer

The sequence is \(32, -8, 2, -\frac{1}{2}, \frac{1}{8}, -\frac{1}{32}, \frac{1}{128}, -\frac{1}{512}\). The formula is \(a_n = 32 \cdot \left(-\frac{1}{4}\right)^{n-1}\).

Step 1: Identify the First Term and Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous one by a constant called the common ratio.The first term of the sequence is \(a_1 = 32\). To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{-8}{32} = -\frac{1}{4}\).

Step 2: Calculate the Next Three Terms

The formula for the \(n\)-th term of a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\).We need to find additional terms: - For \(a_6\): \(a_6 = 32 \cdot \left(-\frac{1}{4}\right)^{5} = 32 \cdot \left(-\frac{1}{1024}\right) = -\frac{1}{32}\)- For \(a_7\): \(a_7 = 32 \cdot \left(-\frac{1}{4}\right)^{6} = 32 \cdot \left(\frac{1}{4096}\right) = \frac{1}{128}\)- For \(a_8\): \(a_8 = 32 \cdot \left(-\frac{1}{4}\right)^{7} = 32 \cdot \left(-\frac{1}{16384}\right) = -\frac{1}{512}\)

Step 3: Numerical Representation

To construct the numerical representation, list the first eight terms of the sequence: \[32, -8, 2, -\frac{1}{2}, \frac{1}{8}, -\frac{1}{32}, \frac{1}{128}, -\frac{1}{512}\]

Step 4: Graphical Representation

Plot the consecutive terms on a graph, taking the term number as the x-axis and the term value as the y-axis. Connect the points to visualize how the term values change with each new term. The term values oscillate and diminish in magnitude as you move along the sequence.

Step 5: Symbolic Representation

The symbolic representation of the sequence is given by the formula for the \(n\)-th term: \[ a_n = 32 \cdot \left(-\frac{1}{4}\right)^{n-1} \]This formula can be used to calculate any term in the sequence by substituting \(n\) with the desired term number.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a critical factor. It determines how each term in the sequence relates to the one before it. For the given sequence, the first term is 32. To find the common ratio, denoted as \(r\), we divide the second term by the first term: \(-8 \div 32 = -\frac{1}{4}\).
This means each term is obtained by multiplying the previous one by \(-\frac{1}{4}\). The common ratio provides a consistent way to understand how a sequence progresses. In this case, the alternate signs and decreasing magnitude are driven by the negative and fractional nature of the common ratio.
It's important to remember that a common ratio can be any real number, and changing its value alters the sequence's behavior drastically. For example, a positive ratio would prevent alternating signs, while a larger absolute value would cause faster increases or decreases in term magnitude.

Graphical Representation

A graphical representation helps visualize a geometric sequence's behavior.
By plotting the term number against its value, you can observe patterns that might not be obvious from the numerical representation alone. For the sequence \(32, -8, 2, -\frac{1}{2}, \frac{1}{8}, -\frac{1}{32}, \frac{1}{128}, -\frac{1}{512}\), you'll notice a couple of things. Each subsequent point gets closer to the x-axis, indicating how the sequence values reduce in magnitude.
The points above and below the axis reflect the alternating positive and negative signs. Connecting these points would reveal an oscillating pattern, a signature of sequences with negative common ratios. Graphical plotting brings a visual element that reinforces the understanding of flow and the alternating nature of the terms.

Symbolic Representation

The symbolic representation of a geometric sequence is a formula that defines any term in pure algebraic terms.
For this sequence, it is expressed as \(a_n = 32 \cdot \left(-\frac{1}{4}\right)^{n-1}\). Understanding and using this symbolic form allows you to calculate any term in the sequence without having to list all previous terms.
The formula uses the first term, 32, and raises the common ratio \(-\frac{1}{4}\) to the power of \(n-1\), where \(n\) is the term number you want to find. This expression encapsulates the sequence's behavior, including its growth pattern and alternating sign.

Numerical Representation

The numerical representation of a sequence offers a straightforward list of the terms. This approach is intuitive and aligns with early stages of mathematical learning.<br> Relating to our exercise, listing the first eight terms becomes simple: \(32, -8, 2, -\frac{1}{2}, \frac{1}{8}, -\frac{1}{32}, \frac{1}{128}, -\frac{1}{512}\). Seeing the numbers directly shows how they diminish in absolute value and alternate in sign.<br> This representation makes it easier to spot patterns visually and helps check calculations if obtained manually via the common ratio or formula.