Problem 38 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=3}^{7}(5 i+2)$$

Step-by-Step Solution

Answer

The sum of the series is 135.

Step 1: Understand the Formula

The series given is a summation from \(i=3\) to \(i=7\) of the expression \(5i+2\). Therefore, the sum can be expressed as: \[ \sum_{i=3}^{7} (5i+2) = (5 \times 3 + 2) + (5 \times 4 + 2) + (5 \times 5 + 2) + (5 \times 6 + 2) + (5 \times 7 + 2) \]

Step 2: Calculate Each Term

Calculate each term in the series separately:- For \(i=3\): \(5 \times 3 + 2 = 15 + 2 = 17\)- For \(i=4\): \(5 \times 4 + 2 = 20 + 2 = 22\)- For \(i=5\): \(5 \times 5 + 2 = 25 + 2 = 27\)- For \(i=6\): \(5 \times 6 + 2 = 30 + 2 = 32\)- For \(i=7\): \(5 \times 7 + 2 = 35 + 2 = 37\)

Step 3: Sum the Terms

Add the results of each term to find the total sum:\[17 + 22 + 27 + 32 + 37 = 135\]

Key Concepts

Understanding the Summation Formula

To solve problems involving series summation, it's crucial to understand the summation notation. This notation allows us to compactly represent the addition of a sequence of terms. The expression \( \sum_{i=3}^{7}(5i+2) \) is an example of using summation notation. Here, \( i \) is called the index of summation, which in this case begins at 3 and ends at 7.
As the variable \( i \) changes from 3 through 7, each value is substituted into the expression \( 5i+2 \) to calculate a term. The summation symbol \( \sum \) is instructing us to sum these calculated terms, starting from the first term (\( i=3 \)) to the last one (\( i=7 \)). Understanding this notation is foundational for correctly evaluating series, as it indicates both the formula for calculating each term and the range of indices to include.
Let's break it down:

• The term after \( \sum \), \( 5i+2 \), is the expression used to generate each element in the series.
• The numbers at the bottom and top of the \( \sum \) indicate the starting (3) and ending values (7) for the index \( i \).
• Plugging in each value from 3 through 7 into \( 5i+2 \) gives the series whose sum we calculate.

Arithmetic Series and Their General Characteristics

Arithmetic series are a type of sequence in which each term increases or decreases by a consistent amount. This amount is called the "common difference." In the context of the arithmetic series, the task is often to sum several terms. Although the given problem \( \sum_{i=3}^{7}(5 i + 2) \) is formulated generally, understanding arithmetic characteristics can still be beneficial.
In arithmetic sequences, if you have a constant common difference between consecutive terms, you can calculate the sum using a specific formula:

• First term: \( a \)
• Common difference: \( d \)
• Number of terms: \( n \)
• Last term: \( l \)

The sum \( S \) of an arithmetic series can be found using the formula: \( S = \frac{n}{2} (a + l) \).
This formula isn't directly used in our problem, but understanding it can help you identify when it's applicable. In our case, the series terms \( 5i+2 \) are generated through substitution into the expression, and individual terms do not form an arithmetic sequence themselves, though some problems may have versions that fit this structure.

Following a Step-by-Step Solution

Step-by-step solutions allow us to systematically approach problems, ensuring no steps are skipped and offering clear guidance. In our problem, the series \( \sum_{i=3}^{7}(5i+2) \) shows how each part of the calculation is tied to its correct sequence.
Let's emphasize what steps to take: **1. Understand the Task:** Identify the series, its range, and the formula term. **2. Calculate Each Term:** Substitute values from 3 to 7 into \( 5i+2 \) one at a time. For example, with \( i=3 \), the calculation becomes \( 5 \times 3 + 2 = 17 \). Continue this until you have each term:

• \( i=3: 17 \)
• \( i=4: 22 \)
• \( i=5: 27 \)
• \( i=6: 32 \)
• \( i=7: 37 \)

**3. Sum the Terms:** Add the individual results together: \( 17 + 22 + 27 + 32 + 37 = 135 \). This is the total sum of the series.
Doing calculations step-by-step ensures accuracy. Skipping over or incorrectly performing any operation can lead to errors, so always check as you go.