Problem 51 - A Graphical Approach to College Algebra

Question

Use a formula to find the sum of each arithmetic series. $$3+5+7+9+11+13+15+17$$

Step-by-Step Solution

Answer

The sum of the arithmetic series is 80.

Step 1: Identify the Sequence Characteristics

This sequence is an arithmetic sequence because the difference between consecutive terms is constant. Here, the first term \( a = 3 \) and the common difference \( d = 5 - 3 = 2 \).

Step 2: Find the Number of Terms

Each term in the sequence can be expressed as \( a_n = a + (n-1) \times d \). To find the number of terms \( n \), use the final term in the sequence: \( 17 = 3 + (n-1) \times 2 \). Solving for \( n \), we have \( 17 - 3 = 2(n-1) \), which simplifies to \( 14 = 2(n-1) \). Dividing by 2 gives \( 7 = n - 1 \), so \( n = 8 \).

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

Use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} \times (a + l) \]where \( a \) is the first term, \( l \) is the last term, and \( n \) is the number of terms. In this case, \( n = 8 \), \( a = 3 \), and \( l = 17 \).

Step 4: Calculate the Sum

Substitute the known values into the summation formula: \[ S_8 = \frac{8}{2} \times (3 + 17) = 4 \times 20 = 80 \]. This calculation gives the sum of the series.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. This is known as the common difference. In this specific example, the sequence is 3, 5, 7, 9, 11, 13, 15, 17. Each number follows the one before it, increasing by the same amount, which makes it an arithmetic sequence.

To easily identify an arithmetic sequence:

• Check if there's a constant difference between consecutive terms.
• Ensure each term increases or decreases by the same amount.

In this exercise, the sequence starts at 3 and increases by 2 each time. After verifying this, you can confidently say that the sequence is arithmetic.

Common Difference

The common difference is a key characteristic of an arithmetic sequence. It's the consistent interval between consecutive terms. In our series, the common difference is calculated by subtracting the first term from the second term: 5 - 3 = 2.

To determine the common difference:

• Identify any two consecutive terms in the sequence.
• Subtract the previous term from the next one.

A correctly identified common difference ensures that all further calculations related to the sequence, like predicting new terms or finding the sum, are accurate.

Sum Formula

The sum of an arithmetic series can be easily calculated using a specific formula. This formula helps you add all terms in the series without needing to add each one individually.

The formula for the sum of the first \( n \) terms in an arithmetic sequence is: \[ S_n = \frac{n}{2} \times (a + l) \] Where:

• \( S_n \) is the sum of the series.
• \( n \) is the number of terms.
• \( a \) is the first term.
• \( l \) is the last term.

For our example: with \( n = 8 \), \( a = 3 \), and \( l = 17 \), plug these values into the formula to find that \( S_8 = 80 \). This method saves time and ensures accuracy in finding the sum of the series.

Number of Terms

Understanding how to determine the number of terms in an arithmetic sequence is crucial. This value is significant because it affects the sum calculation directly. In our sequence example, the term formula \( a_n = a + (n-1) \times d \) helps us find the number of terms.

Steps to find the number of terms:

• Use the last term of the sequence as \( a_n \).
• Insert it into the formula along with the first term \( a \) and common difference \( d \).
• Solve for \( n \) to find out how many terms there are.

In this case, with the sequence closing at 17, substituting into the formula clarifies that there are 8 terms in the series. Knowing this number allows the sum formula to be used effectively, ensuring that your solution is complete and precise.