Problem 20 - Algebra 2 and Trigonometry

Question

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{k=2}^{8}(3+k) $$

Step-by-Step Solution

Answer

The series is expressed as \( 5 + 6 + 7 + 8 + 9 + 10 + 11 \) and its sum is 56.

Step 1: Interpret the Given Series

The series given by \( \sum_{k=2}^{8}(3+k) \) represents the sum of a sequence of terms starting with \( k=2 \) and ending with \( k=8 \). Each term in the series can be expressed as \( 3+k \).

Step 2: Write the Terms of the Series

To explicitly write out the terms, substitute values for each \( k \) from 2 to 8 into \( 3+k \): - For \( k=2 \), the term is \( 3+2 = 5 \).- For \( k=3 \), the term is \( 3+3 = 6 \).- For \( k=4 \), the term is \( 3+4 = 7 \).- For \( k=5 \), the term is \( 3+5 = 8 \).- For \( k=6 \), the term is \( 3+6 = 9 \).- For \( k=7 \), the term is \( 3+7 = 10 \).- For \( k=8 \), the term is \( 3+8 = 11 \).

Step 3: Express Series as Sum of Terms

Combine all the terms identified in Step 2 to express the series explicitly: \[ 5 + 6 + 7 + 8 + 9 + 10 + 11 \]

Step 4: Calculate the Sum

Add all the terms together: \( 5 + 6 + 7 + 8 + 9 + 10 + 11 = 56 \).

Key Concepts

Understanding Summation

Summation is the process of adding numbers or expressions together. It's often represented using the sigma notation \(\sum\), which is a compact way of writing long sums. In the context of an arithmetic series, summation involves adding a sequence of numbers according to a specific rule.

• The index of summation, often denoted as \(k\), tells us where to start and end the summation. For example, \(\sum_{k=2}^{8}\) means "add the terms starting from \(k=2\) to \(k=8\)".
• Each term in the sum is derived from a general formula, like \(3 + k\), which defines the pattern of the series.
• The result of the summation gives the total when all terms are added up together.

Understanding the notation and structure of summation allows us to calculate complex series efficiently.

Sequence of Terms in an Arithmetic Series

A sequence of terms is simply an ordered list of numbers. In arithmetic series, each term follows a specific pattern or formula. This pattern is what characterizes the series. In our example, each term is found by adding a constant value to the term index.

• An arithmetic series is a special type of sequence where each term increases or decreases by a constant difference. In our series, the formula \(3 + k\) shows that each term is formed by adding \(3\) to \(k\).
• The sequence of terms in this series begins at the first calculated term when \(k = 2\), and ends when \(k = 8\). So, the terms are \(5, 6, 7, 8, 9, 10,\) and \(11\).
• Identifying the sequence correctly is crucial to ensure all terms are included in the summation process.

Understanding the sequence of terms aids in systematically adding them together.

Working with Explicit Terms

Explicit terms are terms of a series that are fully written out. In sum notation, the general formula \(3 + k\) is chosen to generate each explicit term.

• Instead of leaving terms in an abstract form, explicitly listing them helps in visualization and verification of the series.
• In our specific series, substituting for \(k\) from \(2\) to \(8\), we explicitly list the terms as \(5, 6, 7, 8, 9, 10,\) and \(11\).
• This step is important for confirming that no terms are missed, and it ensures accuracy before performing the final summation.

Making terms explicit provides clarity and sets the foundation for calculating the sum accurately.

Calculate the Sum of the Series

Once all terms have been identified in a sequence, the next step is to add them together. Calculating the sum of an arithmetic series involves adding each explicit term.

• Start by listing all explicit terms sequentially: \(5, 6, 7, 8, 9, 10,\) and \(11\).
• Proceed to add them one by one: \(5 + 6 + 7 + 8 + 9 + 10 + 11\).
• The result of this sum is \(56\).

This straightforward process confirms the total value of an arithmetic series, concluding the summation task.