Problem 39 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the finite geometric series. The first 20 terms of the series defined by \(a_{n}=3(2)^{n-1}\)

Step-by-Step Solution

Answer

The sum of the first 20 terms is 3,145,725.

Step 1: Identify the Components

The formula for the n-th term of the series is given by \(a_{n} = 3 \times 2^{n-1}\). Here, the first term \(a_1 = 3\) and the common ratio \(r = 2\). Since we need to find the sum of the first 20 terms, \(n = 20\).

Step 2: Use the Sum Formula for a Geometric Series

The sum \(S_n\) of the first \(n\) terms of a geometric series is given by the formula:\[ S_n = a_1 \frac{r^n - 1}{r - 1} \] where \(a_1\) is the first term and \(r\) is the common ratio.

Step 3: Substitute the Values into the Formula

Substitute \(a_1 = 3\), \(r = 2\), and \(n = 20\) into the formula to find the sum:\[ S_{20} = 3 \frac{2^{20} - 1}{2 - 1} = 3 \times (2^{20} - 1) \]

Step 4: Calculate \(2^{20}\)

First, calculate \(2^{20}\), which equals 1,048,576.

Step 5: Compute the Sum

Substitute \(2^{20}\) back into the formula:\[ S_{20} = 3 \times (1,048,576 - 1) = 3 \times 1,048,575 \]

Step 6: Solve for Final Sum

Multiply to find the sum: \( S_{20} = 3 \times 1,048,575 = 3,145,725 \).

Key Concepts

Finite Series

A geometric series is a sum of terms that have a constant ratio between consecutive terms. When discussing a **finite series**, we are referring to the collection of terms that end after a certain term. This is opposed to an infinite series, which would go on indefinitely.

In the given problem, we are dealing with the first 20 terms of the series, making it a finite series.
The notion of a finite series helps us understand how to handle a sum that does not continue infinitely, allowing us to compute an exact total by applying the suitable formulas.

• A finite geometric series has a specific number of terms.
• Each term is defined by a formula involving a common ratio.
• We can use a sum formula to find the total of the series.

Through recognizing the series as finite, it becomes manageable, and we tackle it with straightforward calculations.

Sum Formula

To find the sum of a finite geometric series, we use a specific sum formula that consolidates all the terms of the series into a simple equation.

The sum formula for a geometric series is: \[ S_n = a_1 \frac{r^n - 1}{r - 1} \]where:

• \( S_n \) is the sum of the first \( n \) terms
• \( a_1 \) is the first term
• \( r \) is the common ratio

By plugging the first term and the common ratio into this formula, and computing as needed, we get the sum without having to manually add each individual term.
This formula simplifies the often complex task of summing multiple terms, saving your time and reducing the possibility of errors.

Common Ratio

The **common ratio** is the factor that each term of the series is multiplied by to get the subsequent term. In a geometric progression, this ratio is a constant value, meaning it does not change even as the series progresses.

In our problem, the common ratio \( r \) is determined by the formula \( a_n = 2^{n-1} \). This indicates that each successive term is twice the previous one. This constant feature is what allows the series to be termed a geometric progression.

• The common ratio helps define the growth pattern of the series.
• Knowing the common ratio is essential to using the geometric sum formula.

Understanding the concept of the common ratio is key because it directs how the terms in the series relate to each other. It's a foundational component of calculating the series systematically.

Geometric Progression

A **geometric progression** is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio.
This type of sequence is particularly important due to its applications across different fields such as finance, physics, and computer science.

In our given example, the sequence follows the rule \( a_n = 3(2)^{n-1}\), therefore forming a geometric progression.

• The sequence starts with a specific term.
• Each subsequent term is derived by using the common ratio.

This logical structure is what distinguishes geometric progressions and allows them to be analyzed and summed effectively using formulas designed for them. Understanding the geometric progression helps in constructing each term methodically and forms the basis on which the finite geometric series is built.