Problem 49 - Algebra 2

Question

Find the sum of each geometric series. $$ \sum_{n=1}^{20} 3 \cdot 2^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is 3145725.

Step 1: Identify the First Term and Common Ratio

The given geometric series \( \sum_{n=1}^{20} 3 \cdot 2^{n-1} \) can be represented with the first term \( a = 3 \) and common ratio \( r = 2 \).

Step 2: Understand the Formula for the Sum of a Geometric Series

The formula for 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 3: Plug in the Values

Using the formula \( S_n = a \frac{1-r^n}{1-r} \) with \( a = 3 \), \( r = 2 \), and \( n = 20 \), we get:\[ S_{20} = 3 \frac{1 - 2^{20}}{1 - 2} \]This simplifies to:\[ S_{20} = 3 (2^{20} - 1) \]

Step 4: Calculate the Expression

Now compute \( 2^{20} \):\[ 2^{20} = 1048576 \]Thus, calculate the expression:\[ 3 (2^{20} - 1) = 3 (1048576 - 1) = 3 \cdot 1048575 \]

Step 5: Compute the Final Result

Multiply to find the sum:\[ 3 \cdot 1048575 = 3145725 \]

Key Concepts

Sum of Series

The sum of a geometric series refers to the total value obtained when all terms in the series are added together. A geometric series is a sequence of numbers where each term is derived by multiplying the previous term by a fixed number called the common ratio. The formula for finding the sum of a geometric series is very useful as it simplifies the process significantly, especially for long series.The formula to determine the sum of the first \( n \) terms of a geometric series is:\[S_n = a \frac{1-r^n}{1-r}\]where:

• \( S_n \) is the sum of the series.
• \( a \) is the first term.
• \( r \) is the common ratio.
• \( n \) is the number of terms.

To find the sum, you just need to know the first term, the common ratio, and the number of terms. Plug these values into the formula and perform the calculation. This approach quickly gives the total sum without manually adding each term, which is especially handy for large \( n \).

Common Ratio

In a geometric series, the common ratio is the factor that defines the relationship between consecutive terms. Each term is the product of the previous term and this constant multiplier, known as the common ratio.For example, in the series \( 3, 6, 12, 24, \ldots \), each term increases by a factor of 2. Therefore, the common ratio \( r \) is 2.This constant is crucial because it influences the behavior of the series, whether it converges or diverges as more terms are added:

• If \( |r| < 1 \), the series converges.
• If \( |r| > 1 \), the series diverges.

In our specific example with the geometric series \( \sum_{n=1}^{20} 3 \cdot 2^{n-1} \), the common ratio is \( r = 2 \), indicating that each term is double the previous one.

First Term

In any geometric series, the first term is the initial starting point of the sequence. It's crucial because the value of the first term, combined with the common ratio, determines all the subsequent terms of the series.Taking our geometric series example of \( \sum_{n=1}^{20} 3 \cdot 2^{n-1} \), the first term \( a \) is 3. This is because when \( n = 1 \), the expression \( 3 \cdot 2^{1-1} = 3 \cdot 2^0 = 3 \) clearly shows the first term.Understanding the first term's role helps to set the stage for evaluating the entire series:- It reflects the starting/blocking value of the sequence.- Along with the common ratio, it can precisely predict the behavior of the series.- It's part of the essential components required to use the sum of the series formula.