Problem 7 - Algebra 2

Question

Each sequence has eight terms. Evaluate each related series. $$ \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots, \frac{15}{2} $$

Step-by-Step Solution

Answer

The sum of the series is 32.

Step 1: Identify the First and the Last Term

The first term \(a_1\) in this series is \(\frac{1}{2}\), and the last term \(a_n\) is \(\frac{15}{2}\). The number of terms \(n\) in this series are specified to be 8.

Step 2: Apply the Sum of Arithmetic Series Formula

The formula for the sum \(S_n\) of an arithmetic series is defined as \(S_n = \frac{n}{2}*(a_1 + a_n)\). Substituting \(n = 8\), \(a_1 = \frac{1}{2}\), and \(a_n = \frac{15}{2}\) into the formula, we will get: \(S_8 = \frac{8}{2} * (\frac{1}{2} + \frac{15}{2})\)

Step 3: Evaluate the Sum

After inserting the values into the formula, we simplify to obtain the sum: \(S_8 = 4 * (8) = 32\)

Key Concepts

Sum of Arithmetic Series Formula

An arithmetic series is simply the sum of the terms of an arithmetic sequence. To find the sum of this series, we use the Sum of Arithmetic Series Formula. This formula allows you to quickly calculate the sum without adding each term individually. The formula is:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
Here, \( S_n \) represents the sum of the series, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term of the sequence. By plugging the known values into this formula, you can efficiently find the sum of a series.

First and Last Term Identification

Identifying the first and last terms of the sequence is an essential first step to solving any arithmetic series problem.

• First Term \(a_1\): This is the initial term in the sequence. In our example, \(a_1 = \frac{1}{2}\).
• Last Term \(a_n\): This represents the final term of the sequence. For the given series, \(a_n = \frac{15}{2}\).
• Number of Terms \(n\): Always ensure to verify the total count of terms, which here is 8.

Recognizing these components sets the stage for correctly applying the arithmetic series formula. It helps in streamlining the calculation process, reducing potential errors.

Sequence Evaluation

Once you have used the sum formula, the actual computation of the arithmetic series becomes straightforward. Let's walk through the steps again for clarity.
We know:

• \(n = 8\) - the total number of terms.
• \(a_1 = \frac{1}{2}\) - the value of the first term.
• \(a_n = \frac{15}{2}\) - the value of the last term.

Substitute these values into the formula:
\[S_8 = \frac{8}{2} \times \left( \frac{1}{2} + \frac{15}{2} \right)\]
This simplifies to:
\[S_8 = 4 \times 8 = 32\]
Thus, the sum of the arithmetic sequence is 32. By following these steps carefully, evaluating any given arithmetic series becomes a simple task.