Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=5}^{10}(3 n-3) $$

Step-by-Step Solution

Answer

The sum of the series is 117.

Step 1: Write the General Term Expression

The given arithmetic series is \( \sum_{n=5}^{10}(3n-3) \). The general term of the series is \( a_n = 3n - 3 \). This represents each term in the series as \( n \) takes values from 5 to 10.

Step 2: List Each Term of the Series

Substitute each integer value of \( n \) from 5 to 10 into the expression \( 3n - 3 \) to find each term of the series. - When \( n = 5 \), \( 3(5) - 3 = 12 \) - When \( n = 6 \), \( 3(6) - 3 = 15 \) - When \( n = 7 \), \( 3(7) - 3 = 18 \) - When \( n = 8 \), \( 3(8) - 3 = 21 \) - When \( n = 9 \), \( 3(9) - 3 = 24 \) - When \( n = 10 \), \( 3(10) - 3 = 27 \)

Step 3: Write the Series as a Sum of Terms

Now write the series as a sum of these calculated terms: \[ 12 + 15 + 18 + 21 + 24 + 27 \]

Step 4: Calculate the Sum of the Series

Add all the terms together to find the sum of the series: \[ 12 + 15 + 18 + 21 + 24 + 27 = 117 \]

Key Concepts

Sum of Series

Understanding the sum of an arithmetic series is a fundamental concept in mathematics. An arithmetic series is simply the sum of the terms in an arithmetic sequence, where each term is derived by adding a constant difference to the previous one. In general, the sum of a finite arithmetic series, which includes a specified number of terms, can also be calculated using a specific formula. However, in our exercise, we are focusing on adding up each term individually. This approach is helpful to fully visualize the pattern and ensure every detail is accounted for.

General Term Expression

The general term expression of an arithmetic series serves as a blueprint for all the terms that constitute the series. In our example, the general term is expressed by the formula: - \( a_n = 3n - 3 \) This means any term in the series can be found by substituting the number representing the term's specific position (denoted by \( n \)) into this expression.

• For example, if \( n = 5 \), the term is \( 3 \times 5 - 3 = 12 \).
• This process continues up to \( n = 10 \), showcasing the linear pattern characterized by successive terms.

Substitution in Series

Substitution plays a central role in identifying and writing out each term of the series. By systematically replacing \( n \) with each integer in the defined range, we ensure every term is accounted for. The range in this series spans integers from 5 to 10.

• Start with \( n = 5 \), and substitute into the expression \( 3n - 3 \) to get 12.
• Continue substituting each subsequent integer until you reach the last term. For \( n = 10 \), the term becomes 27.
• Each substitution effectively generates the series' terms: 12, 15, 18, 21, 24, and 27.

Substitution is straightforward but must be done carefully to ensure accuracy.

Calculating Series Sum

Calculating the sum of the series is the culmination of all previous steps. Once all terms are identified, simply adding them together yields the overall sum. - Presented as a sum: \[ 12 + 15 + 18 + 21 + 24 + 27 \] - Add them up step by step, or use a calculator for accuracy: \[ 12 + 15 = 27 \], \[ 27 + 18 = 45 \], and so forth. - Ultimately, the sum is \( 117 \). Practicing this calculation strengthens confidence in handling arithmetic series and ensures a comprehensive grasp of adding sequences directly.