Problem 52 - A Graphical Approach to College Algebra

Question

Use a formula to find the sum of each arithmetic series. $$7.5+6+4.5+3+1.5+0+(-1.5)$$

Step-by-Step Solution

Answer

The sum of the series is 21.

Step 1: Identify the Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. In the series 7.5, 6, 4.5, 3, 1.5, 0, -1.5, you can see that the common difference, \(d\), is -1.5 (e.g., 6 - 7.5 = -1.5).

Step 2: Determine the Number of Terms (n)

To determine the number of terms in the series, count each number. The series has 7 terms.

Step 3: Find the First term (a) and the Last term (l)

The first term \(a\) of the series is 7.5, and the last term \(l\) is -1.5.

Step 4: Use the Formula for Sum of Arithmetic Series

The formula to find the sum \(S_n\) of an arithmetic series is \(S_n = \frac{n}{2} imes (a + l)\).

Step 5: Substitute the Values into the Formula

Substitute the values found into the formula: \(S_7 = \frac{7}{2} imes (7.5 + (-1.5))\).

Step 6: Calculate the Sum

First, calculate \(7.5 + (-1.5)\), which is 6. Then, \(S_7 = \frac{7}{2} imes 6 = \frac{42}{2} = 21\). The sum of the series is 21.

Key Concepts

Common Difference

In an arithmetic series, the common difference is a crucial concept that helps to identify the consistent pattern within the series. It is the amount that each term increases or decreases by, relative to the previous term. Consider the series given in the exercise: 7.5, 6, 4.5, 3, 1.5, 0, -1.5.
To find the common difference, subtract the first term from the second term:

• The common difference, denoted as \(d\), is calculated as \(d = 6 - 7.5 = -1.5\).

This means that each term is 1.5 units less than the preceding term. Understanding this difference is key in performing operations, like finding the sum of the series, since it ensures that each term is consistent with the arithmetic pattern.

Number of Terms

The number of terms in an arithmetic series is simply the total count of all terms in the sequence. Knowing how many terms are in the series is essential for applying various formulas, including those used for finding the sum.
In the given series, we can count each term:

• The series '7.5, 6, 4.5, 3, 1.5, 0, -1.5' has a total of 7 terms.

This count is important as it is directly used in the formula for the sum of an arithmetic series. Make sure to always verify the count to avoid errors in calculations.

Sum of Arithmetic Series

The sum of an arithmetic series can be calculated using a specific formula, which sums all the terms together in an efficient way. Rather than adding each term individually, we use the formula:

• \(S_n = \frac{n}{2} \times (a + l)\)

where \(S_n\) represents the sum of the first \(n\) terms, \(a\) is the first term, \(l\) is the last term, and \(n\) is the number of terms.
In the exercise, with 7 terms, a first term of 7.5, and a last term of -1.5, we substitute these values into the formula:

• \(S_7 = \frac{7}{2} \times (7.5 + (-1.5)) = \frac{7}{2} \times 6\)

After calculating, the sum of the series is found to be 21.

First and Last Term

In any arithmetic series, the first and last terms play pivotal roles in determining the series' sum and overall structure. The first term, often denoted as \(a\), sets the starting point of the series, while the last term, \(l\), marks the endpoint.
In the exercise:

• The first term \(a\) is 7.5.
• The last term \(l\) is -1.5.

These terms are essential for the sum formula \(S_n = \frac{n}{2} \times (a + l)\), allowing us to efficiently compute the sum without individually summing each number. By understanding the role of these terms, you gain insight into both the structure and calculation of arithmetic series.