Problem 44 - Algebra 2

Question

Find the sum of each geometric series. \(3+1+\frac{1}{3}+\cdots\) to 7 terms

Step-by-Step Solution

Answer

The sum of the series is approximately 4.5.

Step 1: Identify the First Term

The first term of the geometric series is given as 3.

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{1}{3} \).

Step 3: Apply the Formula for Sum of a Geometric Series

The formula to find the sum of the first \( n \) terms of a geometric series is \( S_n = a \frac{1-r^n}{1-r} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 4: Substitute Values into the Formula

Substitute \( a = 3 \), \( r = \frac{1}{3} \), and \( n = 7 \) into the formula: \[ S_7 = 3 \frac{1-(\frac{1}{3})^7}{1-\frac{1}{3}} \].

Step 5: Calculate the Power of the Common Ratio

Evaluate \( (\frac{1}{3})^7 \) which equals \( \frac{1}{2187} \).

Step 6: Compute the Sum

Substitute the value from step 5 into the formula: \[ S_7 = 3 \cdot \frac{1 - \frac{1}{2187}}{\frac{2}{3}} \]. The expression simplifies to \[ S_7 = 3 \cdot \frac{2186}{2187} \cdot \frac{3}{2} \].

Step 7: Simplify the Expression

Simplify the expression to find the sum: \[ S_7 = \frac{6558}{2187} \cdot \frac{3}{2} \]. After calculation, \[ S_7 = \frac{9837}{2187} \approx 4.5 \].

Key Concepts

Sum of a Geometric Series

The sum of a geometric series refers to the total when adding all terms of the series for a specified number of terms. Such series follow a specific pattern, where each term is derived by multiplying the previous one by a constant called the common ratio. To find the sum of a geometric series, we utilize a formula that simplifies the process, especially when manual addition is impractical.

For a geometric series with the first term as \(a\), common ratio \(r\), and \(n\) terms, the sum \(S_n\) is calculated using the formula:

• \( S_n = a \frac{1-r^n}{1-r} \)

This formula is invaluable because it handles both increasing and decreasing series efficiently. The expression \(1-r^n\) manages the sum of all terms from the starting to the nth, while \(1-r\) adjusts for the effect of recurring multiplication by the common ratio.

By substituting values into this formula, we can quickly identify the series' sum without adding each term independently. Understanding and applying this formula is fundamental when dealing with series with multiple terms.

Common Ratio

The common ratio in a geometric series is the key to defining the entire sequence. It tells us how each term in the series relates to the preceding one. By consistently multiplying the common ratio to each term, you get the next term.

Finding the common ratio \(r\) is straightforward:

• Divide the second term by the first term.

For example, in the series \(3+1+\frac{1}{3}+\cdots\), the common ratio is \(r = \frac{1}{3}\). This indicates that each term is one-third of the previous term, which leads to decreasing values.

Knowing the common ratio also helps in determining how rapidly a series converges or diverges. A common ratio between -1 and 1 will generally make the series converge, while a ratio outside this range may cause it to diverge extensively. Hence, comprehending the common ratio is crucial not only for finding the series sum but also for predicting any term in the series.

First Term

The first term of a geometric series, often labeled as \(a\), plays a critical role as it sets the stage for the entire sequence. It is the starting point from which all subsequent terms are derived.

In the given series \(3+1+\frac{1}{3}+\cdots\), the first term is \(3\). This first term defines the scale of the series and, paired with the common ratio, determines the amplitude and progression of the series.

Understanding the first term is essential because:

• It directly enters the formula to calculate the sum of the series: \( S_n = a \frac{1-r^n}{1-r} \).
• It impacts the series' value at any term since each subsequent term is essentially the first term multiplied by a power of the common ratio.

Thus, accurately identifying the first term is key to resolving series-related problems, ensuring all calculations related to series are valid and accurate.