Problem 38 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the finite geometric series. $$ 0.6+0.3+0.15+0.075+0.0375 $$

Step-by-Step Solution

Answer

The sum of the geometric series is 1.1625.

Step 1: Identify the First Term (a)

To solve the problem, first identify the first term of the series. Here, the first term, denoted as \( a \), is 0.6.

Step 2: Determine the Common Ratio (r)

Next, identify the common ratio \( r \). The common ratio is found by dividing the second term by the first term: \( r = \frac{0.3}{0.6} = 0.5 \).

Step 3: Count the Number of Terms (n)

Count the number of terms in the series. The series has five terms: 0.6, 0.3, 0.15, 0.075, and 0.0375.

Step 4: Use the Geometric Series Sum Formula

The formula for the sum \( S_n \) of a finite geometric series is \( S_n = a \cdot \frac{1 - r^n}{1 - r} \). Here, \( a = 0.6 \), \( r = 0.5 \), and \( n = 5 \).

Step 5: Calculate \( r^n \)

Calculate \( r^n \), which is \( 0.5^5 = 0.03125 \).

Step 6: Plug Values into the Formula

Substitute the variables into the formula: \( S_5 = 0.6 \cdot \frac{1 - 0.03125}{1 - 0.5} \).

Step 7: Perform the Final Calculation

Evaluate the expression: \( S_5 = 0.6 \cdot \frac{0.96875}{0.5} = 0.6 \cdot 1.9375 \).

Step 8: Finish Calculating the Sum

Complete the multiplication: \( 0.6 \cdot 1.9375 = 1.1625 \). Therefore, the sum of the series is 1.1625.

Key Concepts

Finite Series

A finite series is a sequence of numbers that has a specific number of terms. In the context of a geometric series, each term is multiplied by a constant number called the common ratio to get the next term. The key point about a finite series is that it does not go on indefinitely. It has a clear start and a definite end. This makes finding its sum relatively straightforward, as you only need to consider a limited set of terms.

• The series in our exercise consists of five terms: 0.6, 0.3, 0.15, 0.075, and 0.0375.
• Being finite means you can explicitly list out the terms instead of approaching them as an endless sequence.

Understanding that the series is finite is crucial, as this allows us to apply formulas that are designed specifically for finite series. These formulas help in calculating the sum without needing to perform addition repeatedly for each term.

Sum Formula

To find the sum of a finite geometric series, we can use a specific mathematical expression known as the geometric series sum formula. This formula allows us to calculate the total sum by simply substituting in known values.The formula is:\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \]Here, \( S_n \) is the sum of the series, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

• This formula is powerful because it condenses the series into a simple expression, which saves both time and effort.
• For the series 0.6, 0.3, 0.15, 0.075, and 0.0375, using this formula gives us the sum directly without adding each term individually.

By understanding and applying this formula, you can efficiently find the sum of any finite geometric series, provided you have the necessary values like the first term, common ratio, and number of terms.

Common Ratio

The common ratio in a geometric series is the constant factor you multiply by to get from one term to the next. This ratio is essential because it defines the multiplicative spread between the terms.For example, the exercise problem gives:\[ r = \frac{0.3}{0.6} = 0.5 \]

• This means each term is half the previous term. Understanding this helps make sense of how the series shrinks with each subsequent term.
• The common ratio can be a fraction (as in this case), a whole number, or even a negative number depending on the nature of the geometric series.

Recognizing the common ratio is key to applying the series sum formula correctly, as it directly influences the value of \( r^n \), an important part of the calculation. This gives you insight into the behavior and trend of the series across its terms.

Terms in a Series

When we refer to the terms in a series, we're talking about the individual numbers or elements that make up the sequence. In a geometric series, these terms follow a specific pattern determined by multiplying each term by the common ratio to get the following term.

• In the given series, the terms are 0.6, 0.3, 0.15, 0.075, and 0.0375.
• The number of terms, denoted as \( n \), is critical in calculating the sum of the series.
• Counting the terms correctly ensures you apply the formula accurately.

By analyzing the series' terms, you can verify the pattern and consistency of the geometric progression, ensuring that any calculations follow the correct sequence. Recognizing each term in the series helps confirm the calculations made with the geometric sum formula, tying everything together neatly to arrive at the correct solution.