Problem 40 - 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. \(\sum_{k=1}^{9} 2^{k-1}\) \(\begin{array}{ll}-2-10-50-250 \ldots & 0.4-2+10-50 \ldots\end{array}\)

Step-by-Step Solution

Answer

The sum of the series \(\sum_{k=1}^{9} 2^{k-1}\) is 511.

Step 1: Identify the Series and Formula

The problem involves a geometric series with terms given by \(a_k = 2^{k-1}\). The formula for the sum of the first \(n\) terms of a geometric series \(S_n\) is \(S_n = a_1 \frac{1-r^n}{1-r}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 2: Calculate the First Term \(a_1\)

For the series \(\sum_{k=1}^{9} 2^{k-1}\), the first term when \(k=1\) is \(a_1 = 2^{1-1} = 1\).

Step 3: Determine the Common Ratio \(r\)

In this geometric series, each term is derived by multiplying the previous term by 2, thus \(r = 2\).

Step 4: Use the Formula to Calculate \(S_9\)

Now, use the formula. The series has 9 terms, so \(n = 9\). Plug these into the formula: \[S_9 = 1 \frac{1-2^9}{1-2} = 1 \frac{1-512}{1-2} = \frac{-511}{-1} = 511.\]

Step 5: Verify the Result by Summation

Manually verify by summing the first few terms: 1, 2, 4, 8, ... which doubles each time, confirming the sum calculation from the formula.

Key Concepts

Sum of Series Formula

The sum of a geometric series can be effortlessly calculated using a specific formula. This formula is a powerful tool for finding the total sum of a series of numbers where each term is a fixed multiple of the previous one.
For any geometric series, the formula to find the sum of the first \( n \) terms is given by:

• \[ S_n = a_1 \frac{1-r^n}{1-r} \]

Here,

• \( S_n \) is the sum of the first \( n \) terms.
• \( a_1 \) is the first term of the series.
• \( r \) is the common ratio, which tells us how each term in the series relates to its predecessor.
• \( n \) is the number of terms to sum.

This formula is used to simplify the process of finding the sum without manually adding each term. It saves time and reduces calculation errors in practice.

Common Ratio

In a geometric series, the common ratio \( r \) is a crucial element. It defines the relationship between successive terms. The common ratio is calculated by dividing any term in the series by the preceding term.
For instance, consider a geometric series where the terms are generated using powers of 2: \( 2, 4, 8, 16, \ldots \). Here:

• Each term is obtained by multiplying the previous term by 2.
• Common ratio \( r = 2 \).

The concept of the common ratio is essential because it simplifies the understanding and calculation of subsequent terms in the series. Knowing the common ratio allows you to quickly determine any term or the sum of terms in the series.

Partial Sum Calculation

When dealing with partial sums in a geometric series, the task is to find the sum of a specified number of terms, not necessarily the entire series. This is where the sum of series formula becomes very handy.
For the series \( \sum_{k=1}^{9} 2^{k-1} \), the first term \( a_1 \) is 1, and the common ratio \( r \) is 2. We aim to find the sum of the first 9 terms, hence \( n = 9 \).
The application of our formula goes as follows:

• Plug in \( a_1 = 1 \), \( r = 2 \), and \( n = 9 \) into the formula.
• The result is: \[ S_9 = 1 \frac{1 - 2^9}{1 - 2} = 1 \frac{1 - 512}{-1} = 511 \]

Using the formula simplifies this process, transforming a potentially cumbersome arithmetic task into a straightforward algebraic calculation.

First Term Determination

Identifying the first term of a geometric series, denoted as \( a_1 \), is a straightforward but necessary step because it sets the starting point for every further calculation in the series.
For the series given, \( a_k = 2^{k-1} \), find \( a_1 \) by setting \( k = 1 \):

• \( a_1 = 2^{1-1} = 2^0 = 1 \).

Determining this first term is essential because it directly plugs into our series sum formula. It gives us the foundational quantity from which the series builds. Understanding and identifying \( a_1 \) correctly enables accurate calculations and helps verify each subsequent step in the process.