Problem 47 - Algebra 2

Question

Find the sum of each geometric series. \(\frac{1}{9}-\frac{1}{3}+1-\cdots\) to 6 terms

Step-by-Step Solution

Answer

The sum of the series is \(-\frac{182}{9}\).

Step 1: Identify the First Term and Common Ratio

The given series is a geometric series where the first term \( a \) is \( \frac{1}{9} \) and the second term is \( -\frac{1}{3} \). The common ratio \( r \) is found by dividing the second term by the first term: \( r = \frac{-\frac{1}{3}}{\frac{1}{9}} = -3 \).

Step 2: Use the Formula for the Sum of a Finite Geometric Series

The sum \( S_n \) of the first \( n \) terms of a geometric series can be calculated using the formula: \[ S_n = a \frac{1-r^n}{1-r} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 3: Substitute the Known Values into the Formula

We know \( a = \frac{1}{9} \), \( r = -3 \), and \( n = 6 \). Substitute these into the formula: \[ S_6 = \frac{1}{9} \frac{1 - (-3)^6}{1 - (-3)} \]

Step 4: Calculate \( r^n \) and Other Terms

First, calculate \( (-3)^6 \), which is \( 729 \). Now, substitute back into the sum equation: \[ S_6 = \frac{1}{9} \frac{1 - 729}{1 + 3} \]

Step 5: Simplify the Expression

Simplify inside the fraction: \( 1 - 729 = -728 \) and \( 1 + 3 = 4 \). Substitute these values: \[ S_6 = \frac{1}{9} \frac{-728}{4} \]

Step 6: Final Calculation

Simplify further: \( \frac{-728}{4} = -182 \). Thus, the expression becomes: \[ S_6 = \frac{1}{9} \times -182 = -\frac{182}{9} \]. Therefore, the sum of the series is \(-\frac{182}{9}\).

Key Concepts

Common Ratio

When working with geometric series, understanding the common ratio is crucial. The common ratio, often denoted as \( r \), is the factor by which each term in the series is multiplied to get the next term. Identifying the common ratio helps us understand the pattern of the series and is essential for calculating the sum of the series.

To find the common ratio, divide any term in the series by its preceding term. In our example, the first term is \( \frac{1}{9} \) and the second term is \( -\frac{1}{3} \). Thus, the common ratio is calculated as follows:

• \( r = \frac{-\frac{1}{3}}{\frac{1}{9}} \)
• This simplifies to \( r = -3 \)

Having identified \( r \) as \(-3\), we know that each term is three times larger in magnitude with opposite sign to the previous term.

Sum of Finite Geometric Series

Calculating the sum of a finite geometric series involves a specific formula designed to simplify the process. For the first \( n \) terms of a geometric series, the sum \( S_n \) is given by:

• \( S_n = a \frac{1 - r^n}{1-r} \)

Here, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms you're summing.

The formula for the sum harnesses both the first term and the total number of terms to quickly compute the series' sum without having to add up each term individually, which can be particularly useful when dealing with large series. In our specific example:

• The first term \( a = \frac{1}{9} \)
• The common ratio \( r = -3 \)
• The number of terms \( n = 6 \)

Substituting these values yields:

• \( S_6 = \frac{1}{9} \frac{1 - (-3)^6}{1 - (-3)} \)

By calculating \((-3)^6\) and simplifying the equation, we can compute the series' sum.

First Term

The first term of a geometric series, typically denoted as \( a \), serves as the starting point of the series. It is crucial because all calculations concerning the sum of the series begin with this value.

In any geometric progression, knowing the initial term allows us to predict the subsequent terms when paired with the common ratio. Additionally, when using the sum formula of a geometric series, the first term is a key component within the equation.

In our worked example, the initial term \( a \) is clearly provided as \( \frac{1}{9} \). This is the first number in the sequence from which we calculate the rest of the series. By using the first term in conjunction with the common ratio, we accurately apply the series' sum formula:

• The equation enables us to determine the cumulative value of the series' first six terms.

Identifying the correct first term ensures that all calculations that follow will be based on an accurate foundation, making it indispensable in problems involving geometric series.