Problem 39 - 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. $$-\frac{1}{4},-\frac{1}{2},-1,-2,-4$$

Step-by-Step Solution

Answer

(a) Sequence: \(-\frac{1}{4}, -\frac{1}{2}, -1, -2, -4, -8, -16, -32\). (b) Decreasing exponential graph. (c) Formula: \(a_n = -\frac{1}{4} \cdot 2^{n-1}\).

Step 1: Identify the first term and common ratio

In a geometric sequence, each term is obtained by multiplying the previous term with a common ratio. The first term is given as \(-\frac{1}{4}\). To find the common ratio, divide the second term by the first term: \(-\frac{1}{2} \div -\frac{1}{4} = 2\). Thus, the common ratio is 2.

Step 2: Continue the sequence up to eight terms

Using the first term \(a = -\frac{1}{4}\) and the common ratio \(r = 2\), generate the next terms. Multiply each term by the common ratio:First term: \(-\frac{1}{4}\)Second term: \(-\frac{1}{4} \times 2 = -\frac{1}{2}\)Third term: \(-\frac{1}{2} \times 2 = -1\)Fourth term: \(-1 \times 2 = -2\)Fifth term: \(-2 \times 2 = -4\)Sixth term: \(-4 \times 2 = -8\)Seventh term: \(-8 \times 2 = -16\)Eighth term: \(-16 \times 2 = -32\)These are the first eight terms: \(-\frac{1}{4}, -\frac{1}{2}, -1, -2, -4, -8, -16, -32\).

Step 3: Numerical representation

The first eight terms of the sequence are \(-\frac{1}{4}, -\frac{1}{2}, -1, -2, -4, -8, -16, -32\). This is the numerical representation.

Step 4: Graphical representation

Plot the terms on a graph with the term number on the x-axis and the term value on the y-axis. The sequence is exponentially decreasing, so the points will form a curve that descends rapidly.

Step 5: Symbolic representation

For a geometric sequence, the n-th term can be expressed as \(a_n = a \cdot r^{n-1}\). Substitute \(a = -\frac{1}{4}\) and \(r = 2\) into this formula: \(a_n = -\frac{1}{4} \cdot 2^{n-1}\). This is the symbolic representation of the sequence.

Key Concepts

Numerical Representation

In a geometric sequence, each term is determined by multiplying the previous term by a constant known as the common ratio. For this particular sequence, we've initially been given five terms: \(-\frac{1}{4}, -\frac{1}{2}, -1, -2, -4\). To fully capture the sequence numerically, we extend it to include eight terms, which provides a better understanding of its progression over time.

The process begins with identifying the first term \(a = -\frac{1}{4}\) and the common ratio \(r = 2\). We extend the sequence by repeatedly multiplying each term by 2, resulting in the following numerical representation:

• Sixth term: \(-8\)
• Seventh term: \(-16\)
• Eighth term: \(-32\)

Thus, the first eight terms are \(-\frac{1}{4}, -\frac{1}{2}, -1, -2, -4, -8, -16, -32\). Through this extended list, you can clearly see the exponential pattern governed by the constant multiplication factor.

Graphical Representation

Creating a graphical representation of a geometric sequence can vividly illustrate the nature of its progression. This process involves plotting the terms on a graph, where the x-axis represents the term number and the y-axis represents the term value.

For our specific sequence, the points metaphorically paint a picture of exponential decrease. If you plot the terms \((-1/4, 1), (-1/2, 2), (-1, 3), (-2, 4), (-4, 5), (-8, 6), (-16, 7), (-32, 8)\), you'll notice that the values decline rapidly as you move from left to right along the x-axis.

This visually translates into a steeply declining curve on the graph. Such a graphical representation not only provides insight into how quickly terms shrink but also highlights the significant rate of decay inherent to geometric sequences where the common ratio is greater than 1.

Symbolic Representation

Symbolic representation of a geometric sequence offers a succinct formula that captures the essence of the entire sequence. This representation allows us to compute any term directly, without generating all preceding terms.

The general form for the n-th term of a geometric sequence is expressed as \(a_n = a \cdot r^{n-1}\), where \(a\) is the first term, and \(r\) is the common ratio.

For our sequence, substituting the values gives us \[a_n = -\frac{1}{4} \cdot 2^{n-1}\].

This equation captures the entire behavior of the sequence, allowing for quick calculation of terms regardless of their position, merely by plugging in the desired term number \(n\). This is a powerful tool, turning the seemingly complex nature of sequences into easily manageable expressions.