Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. \(\sum_{n=1}^{10}-2 \cdot\left(\frac{1}{2}\right)^{n-1}\)

Step-by-Step Solution

Answer

The partial sum is \(-\frac{1023}{256}\).

Step 1: Identify the First Term and common ratio

In the series given, the general term \[-2 \cdot \left(\frac{1}{2}\right)^{n-1}\] can be observed. Here, the first term \(a\) is \(-2\), and the common ratio \(r\) is \(\frac{1}{2}\). This is because the general term can be represented as: \(a \cdot r^{n-1}\).

Step 2: Recall the Formula for Partial Sum of a Geometric Series

The formula for the sum of the first \(n\) terms of a geometric series is given by:\[ S_n = a \frac{1-r^n}{1-r} \]where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Here, \(n = 10\).

Step 3: Substitute Values into the Formula

Plug the identified values into the formula:\[ S_{10} = -2 \cdot \frac{1 - \left(\frac{1}{2}\right)^{10}}{1 - \frac{1}{2}} \]This substitution directly uses the first term \(-2\), common ratio \(\frac{1}{2}\), and the number of terms \(10\).

Step 4: Simplify the Expression

First, calculate the power of the ratio:\[ \left(\frac{1}{2}\right)^{10} = \frac{1}{1024} \]Then, the fraction becomes:\[ -2 \cdot \frac{1 - \frac{1}{1024}}{\frac{1}{2}} \]

Step 5: Simplify Further to Get the Result

Simplify the numerator:\[ 1 - \frac{1}{1024} = \frac{1023}{1024} \]Substitute and simplify the whole expression:\[ S_{10} = -2 \cdot \frac{1023}{1024} \cdot 2 = -2 \cdot \frac{1023}{512} \]This simplifies to: \[ S_{10} = -\frac{1023}{256} \]

Key Concepts

Partial Sum

In a geometric series, a partial sum refers to the sum of the first few terms of the series. This is an important concept as it allows you to find the cumulative value of the series up to a certain point without having to sum all the infinite terms. In other words, if you only want to know the total for the first 5 or 10 terms, you'd calculate the partial sum instead of going through all the terms indefinitely.
For a geometric series, like the exercise given, the formula used to determine the partial sum of the first \( n \) terms is:

• \( S_n = a \frac{1 - r^n}{1 - r} \)

Here, \( a \) is the first term, \( r \) is the common ratio, and \( n \) signifies how many terms you want to include in the sum. This formula helps simplify the process by allowing you to directly substitute these values to get the desired sum.

Common Ratio

The common ratio in a geometric progression is a key factor because it describes how the sequence progresses from one term to the next. It tells you by what factor each term multiplies to get to the following term. For example, if the common ratio is \( \frac{1}{2} \), as in the original exercise, then each term is half of the previous term.

• The common ratio, \( r \), can be positive or negative.
• If \( |r| < 1 \), the terms in the series will get smaller over time.
• If \( |r| > 1 \), the terms will grow larger.

To find the common ratio in any sequence, divide any term by the previous term, i.e., \( r = \frac{a_{n+1}}{a_n} \). Understanding the common ratio helps predict the direction and behavior of the series.

Geometric Progression

A geometric progression, or geometric series, is a sequence of numbers where each term after the first is found by multiplying the previous term by the common ratio. This creates a multiplicative pattern across the sequence. Unlike an arithmetic progression, which adds a constant to get from one term to the next, a geometric progression focuses on multiplication.
The series in the example is described by the formula:

• \( a, ar, ar^2, ar^3, \ldots \)

Where \( a \) is the first term and \( r \) is the common ratio. This indicates that, starting from the first term, you keep multiplying by \( r \) for the following terms, which is evident in how the series progresses.

First Term

The first term of a geometric series is what initiates the progression of the sequence. It acts like the starting point from which all subsequent terms are derived by repeated multiplication with the common ratio.
In the given exercise, the first term, \( a \), is \(-2\). This negative first term means the sequence begins in the negative scope, influencing the overall direction of the series.

• The first term can set the "tone" of the entire geometric progression.
• It helps in uniquely defining the particular series you are dealing with.
• To calculate the series using the partial sum formula, the first term is crucial as it backs the entire computation.

Choosing different values for the first term can result in completely different sequences, even if the common ratio remains the same.