Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, use the formula for the sum of the first \(n\) terms of each arithmetic sequence. \(19+25+31+\ldots+73\)

Step-by-Step Solution

Answer

The sum of the arithmetic sequence is 460.

Step 1: Identify the Components of the Sequence

The sequence given is arithmetic, with the first term \(a_1 = 19\) and the last term \(a_n = 73\). The common difference \(d\) can be found by subtracting the first term from the second term: \(d = 25 - 19 = 6\).

Step 2: Determine the Number of Terms

To find the number of terms \(n\) in the sequence, use the formula for the \(n\)-th term of an arithmetic sequence: \[ a_n = a_1 + (n-1) imes d \]Substitute the known values: \[ 73 = 19 + (n-1) imes 6 \]Solve for \(n\):\[ 73 - 19 = (n-1) imes 6 \] \[ 54 = (n-1) imes 6 \] \[ n - 1 = 9 \] \[ n = 10 \] Thus, there are 10 terms in the sequence.

Step 3: Use the Sum Formula for an Arithmetic Sequence

The formula for the sum of the first \(n\) terms of an arithmetic sequence is:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Substitute the known values:\[ S_{10} = \frac{10}{2} (19 + 73) \]\[ S_{10} = 5 \times 92 \]Calculate the sum:\[ S_{10} = 460 \]Thus, the sum of the first 10 terms is 460.

Key Concepts

Sum of Arithmetic Sequence

An arithmetic sequence is a list of numbers where each number after the first is the sum of the previous one plus a constant called the common difference. The sum of the terms in an arithmetic sequence can be calculated easily using a simple formula.To find the sum of an arithmetic sequence, use the formula:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Here:

• \( S_n \) is the sum of the first \( n \) terms.
• \( n \) is the number of terms.
• \( a_1 \) is the first term.
• \( a_n \) is the last term.

In our example, we used this formula to find that the sum of the first 10 terms is 460. This method offers a quick and efficient way to calculate the total of a sequence without adding each term individually.

Common Difference

The common difference is a key concept in an arithmetic sequence. It is the fixed amount each term increases by. To find the common difference, subtract any term from the next term in the sequence.In this exercise, the sequence starts with 19 and then moves to 25. By calculating:\( 25 - 19 = 6 \)we obtained a common difference of 6. This tells us that each subsequent number is 6 more than the previous one. Understanding the common difference helps in identifying the pattern in the sequence and is crucial in calculating other properties of the sequence, such as the number of terms or the sum.

Number of Terms in Sequence

Determining the number of terms in an arithmetic sequence involves using the formula for the \( n \)-th term:\[ a_n = a_1 + (n-1) \times d \]Where:

• \( a_n \) is the last term.
• \( a_1 \) is the first term.
• \( n \) is the number of terms.
• \( d \) is the common difference.

Substituting the values from the problem (such as the first term, last term, and common difference), and solving for \( n \), lets us know how many terms are in the sequence. In this case, when solved correctly, there are 10 terms in the sequence, found by substituting known values into the formula and solving accordingly.

Arithmetic Sequence Formulas

Arithmetic sequences rely heavily on formulas for various calculations. These formulas help find different properties such as specific terms, the total number of terms, and the sum of terms.Key formulas include:- **Nth-term formula**: \( a_n = a_1 + (n-1) \times d \), used to find any term in the sequence.- **Sum formula**: \( S_n = \frac{n}{2} (a_1 + a_n) \), calculates the sum of the first \( n \) terms.Understanding and applying these formulas allows you to quickly find necessary values without manually adding or calculating extensively. In arithmetic sequences, these mathematical tools simplify problem-solving by turning complex problems into simple arithmetic operations.