Problem 35 - College Algebra

Question

Use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum. $$ 6+\frac{15}{2}+9+\frac{21}{2}+12+\frac{27}{2}+15 $$

Step-by-Step Solution

Answer

The sum of the series is 73.5.

Step 1: Identify the Terms

The given series is \( 6, \frac{15}{2}, 9, \frac{21}{2}, 12, \frac{27}{2}, 15 \). These numbers form an arithmetic series.

Step 2: Identify First Term, Last Term, and Number of Terms

The first term \( a \) is 6, and the last term \( l \) is 15. To find the number of terms, we count them: there are 7 terms.

Step 3: Use the Formula for the Sum of an Arithmetic Series

The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by:\[ S_n = \frac{n}{2} (a + l) \]Where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

Step 4: Substitute the Values into the Formula

Substitute \( n = 7 \), \( a = 6 \), and \( l = 15 \) into the formula:\[ S_7 = \frac{7}{2} (6 + 15) \]

Step 5: Calculate the Sum

First calculate \( 6 + 15 = 21 \), then compute \( S_7 = \frac{7}{2} \times 21 \). Calculate \( \frac{7}{2} = 3.5 \) and multiply it by 21 to get \( 73.5 \).

Key Concepts

Sum of Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. To find the sum of the first \( n \) terms of an arithmetic series, we use the formula:

• \[ S_n = \frac{n}{2} (a + l) \]

Here, \( S_n \) represents the sum, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. This formula arises because the series can be thought of as an average of the first and last terms, multiplied by the number of terms.
For example, in the original exercise, to find the sum of the series: 6, \(\frac{15}{2}\), 9, \(\frac{21}{2}\), 12, \(\frac{27}{2}\), 15, we simply insert these values into the formula. Calculating it step-by-step makes understanding easier.

Number of Terms

The number of terms in an arithmetic series refers to how many elements or numbers are included in the sequence. It is denoted by \( n \). Knowing \( n \) is crucial to calculate the total or the sum of the series.
In the given problem, counting the terms directly is the simplest approach, yielding \( n = 7 \). By carefully counting, we include every term between the first and last, confirming the total count needed for calculations.When we use the sum formula, \( n \) directly influences the calculation by dictating how many times the average of the first and last terms is repeated. This shows the integral part \( n \) plays in finding the sum of an arithmetic series.

First Term

The first term of an arithmetic series is simply the initial number in the sequence. It is often denoted by \( a \). The role of the first term is to set the starting point for calculating the sum of the series.
In our exercise, the first term \( a \) is 6. This starting point is crucial since it helps in determining the pattern of the series, and when plugged into the sum formula \( S_n = \frac{n}{2} (a + l) \), it contributes directly to the calculation of the sum. Whether a series increases or decreases from this point is dictated by the common difference, but it is the first term that initiates the sequence, making understanding its value essential in solving arithmetic series problems.

Last Term

The last term of an arithmetic series, denoted by \( l \), is the final number in the sequence. In any arithmetic series, knowing the last term helps complete the calculation of the sum, as it pairs up with the first term in the sum formula.
For the provided series, the last term \( l \) is 15. By knowing this, you can effectively calculate the sum as it pairs with the first term in the averaging process over the number of terms. Plugging \( l \) into the sum formula alongside \( n \) and \( a \) assists in achieving an accurate total.Grasping the significance of the last term is important, especially in ensuring that all elements of the series have been accounted for in the sum calculation.