Problem 18 - College Algebra

Question

Express each geometric sum using summation notation. \(-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\ldots+\frac{1}{768}\)

Step-by-Step Solution

Answer

The series can be expressed as \(\sum_{n=1}^{8} -\frac{1}{6} \left(-\frac{1}{2}\right)^{n-1}\).

Step 1: Identify the First Term and Common Ratio

The series given is \(-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\ldots+\frac{1}{768}\). Identify the first term \(a\) as \(-\frac{1}{6}\). The second term is \(\frac{1}{12}\), so calculate the common ratio \(r\) by dividing the second term by the first: \(r = \frac{1/12}{-1/6} = -\frac{1}{2}\). Thus, \(r = -\frac{1}{2}\).

Step 2: Determine the Pattern of the Series

Notice how the sequence alternates signs and each subsequent term is half of the previous in magnitude, consistent with a geometric series where \(r = -\frac{1}{2}\). Recognize that each term's magnitude halves the preceding term.

Step 3: Express the General Term

Write the general term of the geometric series: \(a_n = a \cdot r^{n-1}\). For this series, it becomes \(a_n = -\frac{1}{6} \cdot \left(-\frac{1}{2}\right)^{n-1}\).

Step 4: Determine the Number of Terms

To express the final term \(\frac{1}{768}\) in terms of the general formula, solve for \(n\) in \(a_n = -\frac{1}{6} \cdot \left(-\frac{1}{2}\right)^{n-1} = \frac{1}{768}\). This involves solving the equation \(-\frac{1}{6} \cdot \left(-\frac{1}{2}\right)^{n-1} = \frac{1}{768}\).

Step 5: Rearrange and Solve for n

Equation rearranges to \(\left(-\frac{1}{2}\right)^{n-1} = \frac{1}{128}\). Since \(-\frac{1}{2}\)^7 equals \(\frac{1}{128}\), \(n-1 = 7\) thus \(n = 8\). So, the geometric series contains 8 terms.

Step 6: Write the Series in Summation Notation

Using all information from above, express the series in summation notation: \(\sum_{n=1}^{8} -\frac{1}{6} \left(-\frac{1}{2}\right)^{n-1}\), which encapsulates all the terms of the series using summation notation.

Key Concepts

Summation Notation

Summation notation is a concise way to write the sum of a sequence of terms. When you're dealing with a geometric series, it allows you to easily express the entire series in a neat and compact form.

To use summation notation, we typically see the Greek letter sigma (\( \Sigma \)) which stands for "sum." To the right, we specify the general term formula of the series, and underneath and on top, we indicate the starting and ending positions in the sequence.

For example, for a geometric series like \(-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\ldots+\frac{1}{768}\), we determined it has 8 terms. In summation notation, we write this as:

• \( \sum_{n=1}^{8} -\frac{1}{6} \left(-\frac{1}{2}\right)^{n-1} \)

This small and elegant expression captures all 8 terms without having to write them out separately.

Common Ratio

The common ratio is a crucial component of geometric series. It's the factor by which we multiply one term to get to the next term in the sequence.

To find the common ratio, we divide any term by the previous term. In our example series \(-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\ldots\), the first term is \(-\frac{1}{6}\), and the second is \(\frac{1}{12}\). We calculate the common ratio \(r\) as follows:

• \( r = \frac{\frac{1}{12}}{-\frac{1}{6}} = -\frac{1}{2} \)

The common ratio informs us about the scaling factor between terms and tells us the nature of the series, whether it is increasing or decreasing, and if the signs alternate.

General Term

In any geometric sequence, you can identify any term given its position in the sequence using the general term formula. The formula for the n-th term in a geometric series is:

• \( a_n = a \cdot r^{n-1} \)

Here, \(a\) is the first term, and \(r\) is the common ratio, and \(n\) corresponds to the term's position in the sequence.

For our specific example series: \(-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\ldots+\frac{1}{768}\), the general term becomes: \(a_n = -\frac{1}{6} \cdot \left(-\frac{1}{2}\right)^{n-1}\).

Using this formula, you can determine not only a specific term but also establish how many terms are in the series.

Geometric Progression

A geometric progression is a series of numbers where each term is found by multiplying the previous term by a constant called the common ratio. It is one of the simplest and most studied types of statistical progression.

The defining feature of geometric progression is how it grows or shrinks exponentially. For our sequence \(-\frac{1}{6},+\frac{1}{12},-\frac{1}{24},\ldots\), every next term is \(-\frac{1}{2}\) times the one before it.

Key characteristics of geometric progressions include:

• Constant ratio between successive terms.
• Can rapidly increase or decrease in size depending on the ratio.
• Can have a minimum or maximum term based on initial conditions.

Understanding geometric progressions lays the groundwork for analyzing how quantities grow or shrink over repeated applications of a consistent rate, which is crucial for fields ranging from finance to physics.