Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=1}^{5}(2 k-2) $$

Step-by-Step Solution

Answer

The sum of the series is 20.

Step 1: Understanding the series

The given series is \( \sum_{k=1}^{5}(2k-2) \). This means we need to evaluate this expression for each value of \( k \) from 1 to 5 and sum the results.

Step 2: Substitute values into the expression

Calculate each term in the series by substituting \( k = 1, 2, 3, 4, 5 \) into the expression \( 2k - 2 \).- Term 1 (\( k = 1 \)): \( 2(1) - 2 = 0 \)- Term 2 (\( k = 2 \)): \( 2(2) - 2 = 2 \)- Term 3 (\( k = 3 \)): \( 2(3) - 2 = 4 \)- Term 4 (\( k = 4 \)): \( 2(4) - 2 = 6 \)- Term 5 (\( k = 5 \)): \( 2(5) - 2 = 8 \)

Step 3: Writing the series as a sum of terms

Now list the results of the substitutions to express the series as an arithmetic sum:\( S = 0 + 2 + 4 + 6 + 8 \).

Step 4: Find the sum

Add the terms calculated:- First, add the first two terms: \( 0 + 2 = 2 \).- Add the result to the next term: \( 2 + 4 = 6 \).- Continue adding: \( 6 + 6 = 12 \).- Lastly, add the final term: \( 12 + 8 = 20 \).Thus, the sum of the series is \( 20 \).

Key Concepts

Sum of Terms

In an arithmetic series, the sum of terms is crucial for understanding the entire sequence. Each term in a series is generated using a specific formula, which helps us find each individual element. Here, the formula is expressed as \(2k - 2\). By substituting integer values sequentially into the formula, we get the terms of the series. Let's break it down:

• For \(k = 1\): \(2(1) - 2 = 0\)
• For \(k = 2\): \(2(2) - 2 = 2\)
• For \(k = 3\): \(2(3) - 2 = 4\)
• For \(k = 4\): \(2(4) - 2 = 6\)
• For \(k = 5\): \(2(5) - 2 = 8\)

By listing these results, we turn the series into a clear addition of terms: \(0 + 2 + 4 + 6 + 8\). This process of breaking down the sequence ensures we completely grasp each component, paving the way for finding the total sum effectively.

Series Evaluation

Series evaluation involves calculating the entire sum from the list of terms. Understanding how to evaluate a series is a fundamental skill. It helps in translating a sequence into a single meaningful value, which represents the series' cumulative impact. Once the individual terms are identified as \(0, 2, 4, 6,\) and \(8\), the next step is to sum them up.

• Start by adding the first two terms: \(0 + 2 = 2\).
• Then, add the next term: \(2 + 4 = 6\).
• Continue this process: \(6 + 6 = 12\).
• Finally, add the last term: \(12 + 8 = 20\).

Thus, the evaluation shows that the sum of the series is \(20\). By performing these additive operations step-by-step, anyone can comprehend how individual terms contribute to the total sum, offering a deeper understanding of the sequence as a whole.

Summation Notation

Summation notation is a concise way to express the sum of a series. Represented by the Greek letter sigma (\(\Sigma\)), this notation allows us to describe lengthy sums more efficiently and effectively. The series in this exercise is originally given in this way as \(\sum_{k=1}^{5}(2k-2)\).The summation notation can be straightforward once you'll understand it:

• \(k\) is the variable, called the "index of summation". It typically starts from the number at the bottom of the sigma, here it's \(1\).
• The upper limit on top of the sigma indicates where the summation stops, here it's \(5\).
• \(2k - 2\) is the expression evaluated at each integer \(k\), generating the terms of the series.

This notation is a powerful tool, simplifying the representation of complex series into compact and manageable forms. Once decoded, it reveals the beauty and pattern within mathematical sequences, making it easier to tackle and solve problems effectively.