Problem 55 - Algebra 2

Question

Use a graphing calculator to find the sum of each geometric series. $$ \sum_{n=1}^{20} 3(-2)^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is -1048575.

Step 1: Identify key elements of the series

This is a geometric series with the first term \( a = 3 \) and common ratio \( r = -2 \). The series goes from \( n = 1 \) to \( n = 20 \).

Step 2: Use the formula for the sum of a geometric series

The formula to find the sum \( S_n \) 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 and \( r \) is the common ratio. Here, \( a = 3 \), \( r = -2 \), and \( n = 20 \).

Step 3: Substitute the values into the formula

Substitute \( a = 3 \), \( r = -2 \), and \( n = 20 \) into the formula: \[ S_n = 3 \frac{1 - (-2)^{20}}{1 - (-2)} \].

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

Using a calculator, calculate \((-2)^{20}\). This result is 1048576.

Step 5: Compute the numerator

Substitute the result from Step 4 into the expression: \[ 1 - (-2)^{20} = 1 - 1048576 = -1048575 \].

Step 6: Compute the denominator

Calculate \(1 - (-2)\) which simplifies to \( 1 + 2 = 3 \).

Step 7: Find the value of the sum

Now substitute the results from Steps 5 and 6 into the sum formula: \[ S_n = 3 \times \frac{-1048575}{3} = -1048575 \].

Step 8: Verify with a graphing calculator

Enter the expression \( \sum_{n=1}^{20} 3(-2)^{n-1} \) into a graphing calculator to confirm the manual calculation. You should obtain the same result.

Key Concepts

Sum of a Series

In mathematics, finding the sum of a series is about adding up all the terms in a sequence. For a geometric series, which is our focus here, the terms are formed by multiplying a constant, known as the _common ratio_, to the previous term. When given a finite geometric series, we can use a specific formula to find its sum easily, without adding each term manually.
The formula is:

• \[ S_n = a \frac{1 - r^n}{1 - r} \]

where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, and \( r \) is the common ratio. This formula is incredibly helpful because it lets us calculate the sum in just a few steps, utilizing basic algebra. By substituting the known values into the formula, you can find the sum of the series quickly and efficiently.

Common Ratio

The common ratio in a geometric series is the value you multiply each term by to get the next term. It defines how quickly or slowly the series grows. For the series \( \sum_{n=1}^{20} 3(-2)^{n-1} \), the common ratio is \( r = -2 \).

• Because it is negative, the sign of each term alternates between positive and negative.
• This alternating pattern affects the series' behavior and makes the sum calculation more interesting.
• A common ratio greater than 1 or less than -1 will cause the series terms to rapidly increase in absolute value.

One crucial step in solving problems involving geometric series is correctly identifying the common ratio, as it is integral to calculating the sum using the geometric series formula.

Graphing Calculator

A graphing calculator is a powerful tool for dealing with complex mathematical computations, like finding the sum of a geometric series. You can use it to verify your manual calculations and explore the series further. Here's what you can do with it:

• Input the series expression directly into the calculator to compute the sum automatically.
• Visualize the trend of the series by plotting its terms.
• Check intermediate results, such as powers and individual terms of the series, to ensure accuracy.

This digital aid not only speeds up calculation but also helps in understanding the underlying patterns of the sequence more vividly. Once you have the formula and parameters ready, using a graphing calculator will save time and serve as a robust confirmation tool for your mathematical processes.

First Term in a Sequence

The first term in a geometric sequence sets the base for all subsequent terms. It's represented by \( a \) in the formula. For our example series \( \sum_{n=1}^{20} 3(-2)^{n-1} \), this first term \( a \) is 3.

• This term is essential because it serves as the starting point of the series.
• All other terms in the series are derived from this initial value by multiplying it by the common ratio multiple times.
• Changing the first term changes all elements of the series since the sequence builds upon it.

Understanding the role of the first term helps in visualizing and calculating the entire series because all consequent terms rely on it. Identifying this correctly ensures the accuracy of the whole problem-solving process.