Problem 9 - Algebra 2

Question

Find the sum of each geometric series. $$ \sum_{n=1}^{8} \frac{1}{3} \cdot 5^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is 32552.

Step 1: Identify the Series Components

This series is a geometric series where each term is given by the formula \(a imes r^{n-1}\). In this case, \(a = \frac{1}{3}\) is the first term of the series, and \(r = 5\) is the common ratio.

Step 2: Determine the Formula for the Sum

The sum \(S_n\) of the first \(n\) terms of a geometric series is given by the formula \(S_n = a \frac{r^n - 1}{r - 1}\). For this series, \(a = \frac{1}{3}\), \(r = 5\), and \(n = 8\).

Step 3: Substitute Values into the Formula

Plug the values \(a = \frac{1}{3}\), \(r = 5\), and \(n = 8\) into the sum formula: \(S_8 = \frac{1}{3} \frac{5^8 - 1}{5 - 1}\). This simplifies to \(S_8 = \frac{1}{3} \frac{390625 - 1}{4}\).

Step 4: Calculate the Exponentiation

First, calculate \(5^8 = 390625\). Subtract 1 to get \(390624\).

Step 5: Simplify the Expression

Now substitute \(390624\) back into the formula: \(S_8 = \frac{1}{3} \frac{390624}{4}\). This simplifies to \(S_8 = \frac{1}{3} \times 97656\).

Step 6: Final Calculation

Multiply \(97656\) by \(\frac{1}{3}\) to get the final sum: \(S_8 = 32552\).

Key Concepts

The Common Ratio

In a geometric series, the common ratio is a crucial element that defines the series' structure. It is the factor by which each subsequent term is multiplied to get the next term in the series. Understanding this ratio helps in identifying the pattern and nature of the series.

• Definition: The common ratio, often denoted as \( r \), is calculated by dividing any term by the previous term in the series.
• Example: In the series given by \( \frac{1}{3} \cdot 5^{n-1} \), the common ratio is \( 5 \). This means each term is 5 times the previous term.
• Importance: The common ratio helps in quickly identifying if a series is geometric and in finding the value of future terms without having to calculate each one individually.

Recognizing the common ratio is the first step in solving and understanding a geometric series. It provides the constant factor that leads to the exponential growth or decay characteristic of geometric sequences.

The First Term

The first term of a geometric series is the initial value or starting point of the sequence. It is vital as it sets the stage for the entire series and plays a significant role in calculating the series' sum.

• Definition: The first term, often represented by \( a \), is simply the first number in the series.
• Example: In the series \( \frac{1}{3} \cdot 5^{n-1} \), the first term is \( \frac{1}{3} \), since when \( n=1 \), the expression simplifies to \( \frac{1}{3} \cdot 5^{0} = \frac{1}{3} \cdot 1 = \frac{1}{3} \).
• Role in Calculations: The first term is crucial in the formula for finding the sum of a geometric series and influences the overall value of the sum.

Understanding the first term allows for the precise calculation of subsequent terms and lays the foundation for deriving the sum of the series.

Sum of Geometric Series

The sum of a geometric series refers to the total when all the terms from the first to the \( n \)-th term are added together. Calculating this sum is a common problem in mathematics, especially when the series includes many terms.

• Formula: The sum \( S_n \) of the first \( n \) terms is given by \( S_n = a \frac{r^n - 1}{r - 1} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
• Application: Using the formula with our series, where \( a = \frac{1}{3} \), \( r = 5 \), and \( n = 8 \), we substitute these values to find: \[ S_8 = \frac{1}{3} \frac{5^8 - 1}{5 - 1} = \frac{1}{3} \times 97656 = 32552 \]
• Importance: Knowing how to find the sum of a geometric series can simplify complex problems in financial calculations, physics, and other areas of science.

Mastering the process of calculating the sum of a geometric series allows one to handle broader mathematical and practical problems with ease.