Problem 39 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=-2}^{3} 2(3)^{i}$$

Step-by-Step Solution

Answer

The sum of the series is approximately 80.89.

Step 1: Understand the Series

The task requires finding the sum of a series given by \( \sum_{i=-2}^{3} 2(3)^{i} \). This is a finite geometric series, where each term is of the form \( 2 \times 3^{i} \), and \( i \) ranges from \( -2 \) to \( 3 \).

Step 2: Identify the First Term and Common Ratio

The first term of the sequence when \( i = -2 \) is \( 2 \times 3^{-2} = \frac{2}{9} \). The common ratio \( r \) can be identified as \( 3 \), because each term is multiplied by 3 as \( i \) increases by 1.

Step 3: Use the Geometric Series Formula

In a finite geometric series, the sum is given by the formula \[ S_n = a \frac{r^{n} - 1}{r - 1} \]where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a = \frac{2}{9} \), \( r = 3 \), and the number of terms \( n = 6 \), corresponding to terms from \( i = -2 \) to \( i = 3 \).

Step 4: Calculate the Number of Terms

Since \( i \) ranges from \( -2 \) to \( 3 \), the total number of terms is \( 3 - (-2) + 1 = 6 \).

Step 5: Substitute into Formula and Simplify

Substitute \( a = \frac{2}{9} \), \( r = 3 \), and \( n = 6 \) into the formula:\[S_6 = \frac{2}{9} \frac{3^6 - 1}{3 - 1} = \frac{2}{9} \frac{729 - 1}{2} = \frac{2}{9} \times 364 = \frac{728}{9} = 80.8888\cdots\]Convert to a simpler fraction or approximately: the sum is 80.89.

Key Concepts

Finite Geometric Series

A finite geometric series is a sequence of numbers with a specific pattern. In this series, each term after the first is found by multiplying the previous one by a fixed number called the common ratio. The term 'finite' indicates that this series has a specific number of terms.
Here’s a quick way to understand it:

• The first term is usually denoted as \( a \).
• Each term is obtained by multiplying the previous term by the common ratio \( r \).
• The series ends after a certain number of terms \( n \).

In the example given, the series begins with \( 2 \times 3^{-2} \) and continues to \( 3^3 \). Calculating each term precisely creates a clear path to find the sum.

Geometric Sequence

A geometric sequence is the foundation upon which a geometric series is built. It consists of terms following a strict rule where each term is the product of the previous term and a constant known as the common ratio \( r \).
Key features include:

• The sequence is defined by its first term \( a \) and the common ratio \( r \).
• Consistency in multiplying the common ratio makes the pattern predictable.
• This sequence is used to form the series, which might be infinite or finite.

In our specific example, starting with the first term \( \frac{2}{9} \) when \( i = -2 \), each subsequent term is found by multiplying by \( 3 \). The progression from start to finish gives rise to the corresponding series.

Sum of Series

The sum of a geometric series is concluded using a specific formula. This allows us to efficiently calculate the total without manually adding each term.
The formula for the sum \( S_n \) of a finite geometric series is:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
where:

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

The sum is derived by substituting these values into the formula, leading to a fast computation. For instance, in the given exercise, substituting \( \frac{2}{9} \) for \( a \), \( 3 \) for \( r \), and \( 6 \) for \( n \) yields:
\[ S_6 = \frac{2}{9} \frac{3^6 - 1}{2} = \frac{728}{9} \approx 80.89 \]This process streamlines the calculation and provides the sum of the series.