Problem 33 - Algebra 2

Question

Find the sum of each geometric series. $$ \sum_{n=1}^{6} 2(-3)^{n-1} $$

Step-by-Step Solution

Answer

The sum of the series is \(-364\).

Step 1: Identify the First Term and Common Ratio

Recognize that this is a geometric series. For the series \( \sum_{n=1}^{6} 2(-3)^{n-1} \), the first term \( a \) is the value of the series at \( n=1 \), which is \( a = 2 \cdot (-3)^{1-1} = 2 \). The common ratio \( r \) is the base inside the exponent, \( r = -3 \).

Step 2: Write the Geometric Series Formula

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

Step 3: Substitute Values into the Formula

Substitute the known values into the formula: \( a = 2 \), \( r = -3 \), and \( n = 6 \). So, the sum becomes: \[ S_6 = 2 \frac{1 - (-3)^6}{1 - (-3)} \]

Step 4: Simplify the Expression

Calculate \((-3)^6\). Since \( (-3)^6 = 729 \), substitute that into the equation: \[ S_6 = 2 \frac{1 - 729}{1 + 3} \] Simplifying further: \[ S_6 = 2 \frac{-728}{4} = 2 \times (-182) \]

Step 5: Calculate the Final Sum

Complete the final multiplication: \[ S_6 = -364 \]. The sum of the series is \(-364\).

Key Concepts

sum of series

The concept of the sum of a series is central to understanding how all the terms in a sequence add up to a single value. In a geometric series, each term is derived by multiplying a constant factor, known as the common ratio, with the previous term. The formula for the sum of the first \( n \) terms of a geometric series is given by:

• \( S_n = a \frac{1 - r^n}{1 - r} \)

In this formula, \( S_n \) represents the sum of the series, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the total number of terms to be added. To find the sum of a geometric series, you substitute the known values into this formula and simplify the expression. This lets you calculate the total of all terms from the first up to the nth term.
In the given example, we used this formula to find the sum of the series up to the sixth term, which resulted in a sum of \(-364\).

common ratio

The term common ratio refers to the constant factor that is multiplied with each term in a geometric sequence to get the next term. It can be either positive or negative, and it significantly affects the series' behavior. If the common ratio \( r \) is greater than 1 or less than -1, the series increases or decreases very quickly.
In our problem, the common ratio is \(-3\). This means, to determine the next term in the series, you multiply the current term by \(-3\). For instance, starting with the first term as 2, the second term would be \(2 \times (-3) = -6\), the third term would be \(-6 \times (-3) = 18\), and so on.
The behavior of a series is determined largely by the common ratio; if it's negative, the series will alternate signs, leading to terms that are positive and negative alternately. Understanding the common ratio helps predict how fast the series grows or how it oscillates as each term is generated.

first term

The first term of a geometric series, often denoted by the letter \( a \), is crucial because it serves as the starting point for the sequence. It determines the value of the initial term before the multiplication by the common ratio begins.
In the example provided, the first term \( a \) was found by evaluating the expression when \( n = 1 \):

• \( a = 2 \cdot (-3)^0 = 2 \)

This initial step is vital as it forms the base from which all other terms are derived using the common ratio. Each subsequent term in the series is a multiplication of this first term with powers of the common ratio. Identifying the first term correctly is essential because any error at this stage will affect the entire sequence and the eventual sum of the series.

geometric sequence

A geometric sequence is a list of numbers where each term after the first is obtained by multiplying the previous term by a constant, known as the common ratio. Unlike arithmetic sequences, where terms increase by addition, a geometric sequence grows or shrinks exponentially.
Exploring the sequence formed by \( ", 2, -6, 18, -54, 162, -486 \) reflects the operation of continuously multiplying by the common ratio \(-3\).
This method of progression distinguishes geometric sequences and makes them useful for modeling various scenarios, such as population growth, decay of radioactive substances, and financial calculations where compound interest applies. Understanding their structure is crucial for solving problems that involve exponential growth or alternating patterns.