Problem 7 - Algebra 2

Question

Evaluate the finite series for the specified number of terms. $$ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots ; n=8 $$

Step-by-Step Solution

Answer

The sum of the first eight terms of the given series is 255.5

Step 1: Identify the values

First, identify the first term \(a\), the common ratio \(r\), and the number of terms \(n\). Here, \(a = \frac{1}{2}\), \(r = \frac{1}{2}\), and \(n = 8\).

Step 2: Substitute values into the formula

Next, substitute these values into the formula for the sum of a finite geometric series \(S_n = \frac{a(1 - r^n)}{1 - r}\). This gives \(S_8 = \frac{\frac{1}{2}(1 - (\frac{1}{2})^8)}{1 - \frac{1}{2}}\).

Step 3: Simplify the expression

Now, simplify the expression to find the series sum. This results in \(S_8 = \frac{(1 - \frac{1}{256})}{\frac{1}{2}} = 255.5\)

Step 4: Interpret Result

The sum of the first 8 terms of the given geometric series is 255.5

Key Concepts

Geometric Sequences: Understanding the Basics

In mathematics, a geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the "common ratio." Consider the sequence \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \). Here, the first term \(a\) is \( \frac{1}{2} \), and the common ratio \(r\) is \( \frac{1}{2} \). This means each term is half of the term before it.

• First term (\(a\)): \( \frac{1}{2} \)
• Common Ratio (\(r\)): \( \frac{1}{2} \)

Understanding the basic structure of a geometric sequence is essential because it sets the foundation for solving problems involving series. Once you identify \(a\) and \(r\), you can start thinking about the series that these sequences form.
It's crucial to be comfortable with determining these values, as they are used in further calculations for series sums.

Sum of Series Formula: Calculating the Total

Once you've identified the components of a geometric sequence, such as the first term \(a\), the common ratio \(r\), and the number of terms \(n\), you can calculate the sum of a finite series using a specific formula. The formula for the sum \(S_n\) of the first \(n\) terms in a geometric series is:\[S_n = \frac{a(1 - r^n)}{1 - r}\]
To apply this formula:

• Insert the value of \(a\).
• Raise the common ratio \(r\) to the power of \(n\).
• Plug these into the formula to find the series sum.

For instance, for our sequence \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \) with \(n = 8\), the sum \(S_8\) is calculated by inserting the known values:\[\frac{\frac{1}{2}(1 - (\frac{1}{2})^8)}{1 - \frac{1}{2}}\]
This formula is powerful because it calculates the total sum without listing or individually adding all the terms, especially as the sequence grows longer.

Series Simplification: Breaking Down Expressions

After you have applied the sum of series formula, simplifying the mathematical expression is your next step. This means you need to perform basic arithmetic and algebraic manipulations to get an explicit result. In the example \[S_8 = \frac{\frac{1}{2}(1 - (\frac{1}{2})^8)}{1 - \frac{1}{2}}\], follow these steps to simplify:

• Calculate \((\frac{1}{2})^8\), resulting in \(\frac{1}{256}\).
• Subtract \(\frac{1}{256}\) from 1, yielding \(\frac{255}{256}\).
• Multiply this by \(\frac{1}{2}\).
• Finally, divide by \(\frac{1}{2}\) to find the sum.

This refinement yields a more transparent result of 255.5, which is the sum of the first 8 terms of the sequence. Simplifying is crucial as it gives a clear numeric answer and helps in understanding the series' behavior by showing how each component contributes to the final sum.