Problem 39 - College Algebra

Question

Use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. \(S_{7}\) for the series \(0.4-2+10-50 \ldots\)

Step-by-Step Solution

Answer

The partial sum \(S_7\) is 5208.4.

Step 1: Identify the first term and common ratio

The first term of the series is represented by \(a\), which is the initial number in the series. Here, \(a = 0.4\). The common ratio \(r\) can be determined by dividing the second term by the first term. So, \(r = \frac{-2}{0.4} = -5\).

Step 2: Use the formula for the sum of the first n terms

The formula for the sum of the first \(n\) terms of a geometric series is given by: \[ S_n = a \frac{1-r^n}{1-r} \]Here, \(n = 7\), \(a = 0.4\), and \(r = -5\).

Step 3: Plug values into the formula

Substitute the identified values into the formula:\[ S_7 = 0.4 \frac{1-(-5)^7}{1-(-5)} \]

Step 4: Calculate \(-5^7\)

Compute \((-5)^7\):\((-5)^7 = -78125\).

Step 5: Substitute and simplify the expression

Substitute \(-78125\) back into the formula:\[ S_7 = 0.4 \frac{1 - (-78125)}{1 + 5} \]This simplifies to:\[ S_7 = 0.4 \frac{1 + 78125}{6} \]\[ S_7 = 0.4 \frac{78126}{6} \]

Step 6: Calculate the sum

Divide 78126 by 6:\[ \frac{78126}{6} = 13021 \].Then multiply by 0.4:\[ S_7 = 0.4 \times 13021 = 5208.4 \].

Key Concepts

Sum of Series

A geometric series is a series of terms where each term is obtained by multiplying the previous one by a constant, known as the common ratio. To find the sum of such a series, especially when the number of terms is specified, we use a particular formula. This formula is essential in calculating the sum without the hassle of adding each term individually, especially for large numbers of terms.
For a geometric series, the sum of the first \( n \) terms, denoted as \( S_n \), can be determined using the following formula:

• \( S_n = a \frac{1-r^n}{1-r} \)

Here, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
This formula works effectively when \( r eq 1 \), helping us to quickly find the sum by substituting the known values of \( a \), \( r \), and \( n \).
In practical scenarios, this method simplifies the process, allowing one to focus more on understanding patterns rather than tedious calculations. It provides a quick and efficient way to calculate the sum as demonstrated in the original problem where we computed the partial sum for the first 7 terms of a geometric series.

Common Ratio

Identifying the common ratio is a crucial step in working with geometric series. The common ratio \( r \) defines the factor that each term is multiplied by to get the next term in the series.
To find \( r \), one can simply divide the second term of the series by the first term:

• \( r = \frac{\text{second term}}{\text{first term}} \)

For example, in the series \( 0.4, -2, 10, -50, \ldots \), the first term \( a = 0.4 \), and the second term is -2.
Thus, \( r = \frac{-2}{0.4} = -5 \).
This negative common ratio indicates that the terms will alternate between positive and negative values, increasing in magnitude by a factor of 5 each time.
Understanding the role of the common ratio helps predict the behavior of the series, offering insights into the growth or decay pattern of the sequence, which is vital for correctly applying the sum formula.

Partial Sum

The concept of a partial sum in a geometric series refers to the sum of a specified number of consecutive terms, starting from the first term. Calculating a partial sum is useful when complete information about the series is either unavailable or unnecessary.
Given a geometric series with first term \( a \), common ratio \( r \), and \( n \) terms, the partial sum \( S_n \) is calculated using the sum formula:

• \( S_n = a \frac{1 - r^n}{1 - r} \)

This formula gives us the sum of the first \( n \) terms by taking into account the effects of the common ratio over these terms.
For instance, if we need to find \( S_7 \) for a series like in our example, the formula provides a straightforward method of obtaining this sum by evaluating \( r^7 \), then performing arithmetic operations based on the given formula.
This concept is especially helpful in mathematical tasks where only a partial contribution to the total is needed, such as the problem in the original exercise where only the first 7 terms' sum was required.

First Term

In any geometric series, the first term, denoted \( a \), serves as the starting point for all subsequent terms. Recognizing and correctly identifying this term is crucial as it forms the foundation of the entire series.
In the context of calculating the sum of a series, \( a \) sets the initial value from which the series progresses through multiplication by the common ratio.
In our example series, the first term is given as 0.4. This value is used directly in the formula for the sum of a geometric series:

• \( S_n = a \frac{1 - r^n}{1 - r} \)

The presence of this term in the formula signifies its continuous influence throughout the series, adjusting the scale of the sum based on the value of \( a \).
Without correctly identifying the first term, calculations might lead to inaccurate results, misrepresenting the actual sum of the series or, in the case of partial sums, failing to properly account for their foundational starting point.