Problem 35 - 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. $$8,4,2,1, \frac{1}{2}$$

Step-by-Step Solution

Answer

The sequence is represented numerically as 8, 4, 2, 1, 0.5, 0.25, 0.125, 0.0625. Graphically, it decays exponentially; symbolically, it's \( a_n = 8 \cdot (\frac{1}{2})^{n-1} \).

Step 1: Identify the First Term and Common Ratio

The first term of the sequence \( a_1 \) is clearly given as 8. To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{4}{8} = \frac{1}{2} \). Thus, the common ratio \( r \) is \( \frac{1}{2} \).

Step 2: Calculate Additional Terms

To expand the sequence to at least eight terms, continue multiplying each term by the common ratio \( r \). Thus, the next terms are: \( a_6 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \), \( a_7 = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \), and \( a_8 = \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \). The first eight terms are: 8, 4, 2, 1, \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \), \( \frac{1}{16} \).

Step 3: Numerical Representation

The numerical representation of the sequence is simply the list of the first eight terms: 8, 4, 2, 1, \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \), \( \frac{1}{16} \).

Step 4: Graphical Representation

Plot the terms of the sequence on a graph with the term number on the x-axis and the value of the term on the y-axis. You'll see exponential decay as the values decrease by a factor of 2 with each subsequent term.

Step 5: Symbolic Representation

The symbolic representation of a geometric sequence can be expressed in the form \( a_n = a_1 \cdot r^{(n-1)} \). With \( a_1 = 8 \) and \( r = \frac{1}{2} \), the n-th term is given by \( a_n = 8 \cdot \left(\frac{1}{2}\right)^{n-1} \). This formula can be used to find any term in the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that helps in identifying the relationship between consecutive terms. It is calculated by dividing one term by the preceding term. For instance, in the sequence provided: 8, 4, 2, 1, \( \frac{1}{2} \), the common ratio \( r \) is found as follows:

• Divide the second term by the first term: \( r = \frac{4}{8} = \frac{1}{2} \).

This common ratio \( r = \frac{1}{2} \) explains how the sequence progresses. Each term is half the value of the previous term, showcasing a repetitive pattern of division by 2.
Understanding the common ratio allows for the prediction of future terms and provides insight into the behavior of the sequence over time.

Graphical Representation

Visualizing a geometric sequence can be incredibly helpful, and a graphical representation does just that. By plotting the sequence on a graph, you use the term number on the x-axis and the term value on the y-axis.

• The first eight terms of our sequence are: 8, 4, 2, 1, \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \), \( \frac{1}{16} \).
• As you plot these points, you'll observe a distinct pattern of exponential decay, where each value is reduced by a factor of 2.

This graphical pattern helps visualize how quickly the terms of the sequence decrease. It's a clear representation of the characteristics of a geometric sequence with a common ratio less than 1. Such visual aids can be beneficial for comprehending how values transform over several iterations.

Symbolic Representation

The symbolic representation of a geometric sequence provides a formulaic approach for determining any term within the sequence. For a sequence where the first term \( a_1 \) is known, and the common ratio \( r \) is determined, the n-th term \( a_n \) can be expressed as:

• \( a_n = a_1 \cdot r^{(n-1)} \)
• Here, \( a_1 = 8 \) and \( r = \frac{1}{2} \).

Thus, the formula for the n-th term becomes: \[ a_n = 8 \cdot \left(\frac{1}{2}\right)^{n-1} \] This expression allows you to compute any term number effortlessly by just substituting the desired term number (n) into the formula.
It's a concise representation capturing the essence of the entire sequence compactly.

Numerical Representation

A numerical representation involves listing out the sequence explicitly as individual numbers. For the geometric sequence provided, the first eight terms are:

• 8, 4, 2, 1, \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \), \( \frac{1}{16} \).

This straightforward list highlights the geometric progression pattern, showcasing how each subsequent term is formed by multiplying the previous term by the common ratio \( \frac{1}{2} \).
The numerical approach is helpful for quickly identifying or verifying terms within the sequence as it provides a clear, step-by-step account without any abstractions.