Problem 36 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the finite geometric series. $$ 2+\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\frac{1}{128}+\frac{1}{512} $$

Step-by-Step Solution

Answer

The sum of the series is approximately 2.6640625.

Step 1: Identify the series parameters

We need to identify the first term and the common ratio of the series. The first term \( a \) is 2. To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{1}{2} \div 2 = \frac{1}{4} \).

Step 2: Count the number of terms

Count the number of terms \( n \) in the series. The terms are: 2, \( \frac{1}{2} \), \( \frac{1}{8} \), \( \frac{1}{32} \), \( \frac{1}{128} \), \( \frac{1}{512} \), giving us a total of 6 terms.

Step 3: Use the formula for the sum of a finite geometric series

The formula for the sum of a finite geometric series is \( S_n = a \frac{1-r^n}{1-r} \). Here, \( a = 2 \), \( r = \frac{1}{4} \), and \( n = 6 \).

Step 4: Calculate \( r^n \)

Calculate \( r^n \), which is \( \left(\frac{1}{4}\right)^6 = \frac{1}{4096} \).

Step 5: Substitute the values into the formula

Substitute \( a \), \( r \), and \( r^n \) into \( S_n = 2 \cdot \frac{1 - \frac{1}{4096}}{1 - \frac{1}{4}} \).

Step 6: Simplify the expression

Simplify the expression: \( S_n = 2 \cdot \frac{4095}{4096} \cdot \frac{4}{3} = \frac{8190}{3072} \). After simplifying further, \( S_n = \frac{2730}{1024} = 2.6640625 \).

Key Concepts

Geometric Series

A geometric series is a sequence of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." This type of series is fundamental in college algebra and serves as the basis for understanding many mathematical patterns. The series starts with an initial term, labeled as \( a \), and progresses by repeatedly applying the common ratio \( r \). For example, in the series \( 2 + \frac{1}{2} + \frac{1}{8} + ... \), each term is obtained by multiplying the previous term by \( \frac{1}{4} \). Geometric series have important applications in finance, computer science, and physics because of their predictable growth patterns.

Geometric series can be finite or infinite, but in this exercise, we focus on a finite series.

Finite Series Sum

The sum of a finite series involves adding a specific number of terms in the geometric sequence. To find this sum, we use a special formula. This allows for quick calculations without manually adding each term. The formula for a finite geometric series 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
• \( r \) is the common ratio
• \( n \) is the number of terms

In our exercise, \( a = 2 \), \( r = \frac{1}{4} \), and \( n = 6 \), giving us a clear path to calculate the sum of the given series. Calculating the sum of a finite series is especially useful in business for calculations like loan repayments or determining total sales over a set period.

In solving our problem, we used this formula to find the sum \( S_n = 2.6640625 \).

Common Ratio

The common ratio is the factor that each term is multiplied by to get to the next term in a geometric series. Identifying the common ratio is crucial because it determines how the series grows or shrinks. To find the common ratio, you divide any term in the series by the term preceding it. In our series:

• The first term is 2
• The second term is \( \frac{1}{2} \)

Thus, the common ratio \( r \) is \( \frac{1}{2} \div 2 = \frac{1}{4} \).

The common ratio can tell you if the series is growing (if \( |r| > 1 \)) or shrinking (if \( |r| < 1 \)). In this problem, since \( r = \frac{1}{4} \), the series is decreasing, which affects the sum of the series and shows how quickly the terms reduce.

Algebraic Formula Application

Using algebraic formulas efficiently can simplify solving problems involving complex sequences like geometric series. In this case, we applied the formula for the sum of a finite geometric series. First, we identified the necessary parameters: the first term \( a \), the common ratio \( r \), and the number of terms \( n \). Next, we calculated \( r^n \), which is essential for accurately using the formula.

By substituting \( a = 2 \), \( r = \frac{1}{4} \), and \( r^n = \frac{1}{4096} \) into \( S_n = a \frac{1-r^n}{1-r} \), we were able to efficiently compute the series sum. Algebraic formula application is a powerful tool that transforms problems that seem complicated at first glance into manageable solutions. For students, mastering these formulas enhances problem-solving skills and aids in tackling more advanced mathematical concepts in college algebra.