Problem 35 - Algebra 2

Question

Find the sum of each arithmetic series. $$ 7+14+21+28+\dots+98 $$

Step-by-Step Solution

Answer

The sum of the series is 735.

Step 1: Identify the Parameters of the Series

The given arithmetic series is 7, 14, 21, 28, ..., 98. Identify the first term and the common difference. The first term \( a \) is 7, and the common difference \( d \) is 7 (since 14 - 7 = 7). Now, find the last term \( l \), which is 98.

Step 2: Calculate the Number of Terms

To find the sum, we need to determine the number of terms in the series. Use the formula for the \( n \)-th term of an arithmetic series: \( a_n = a + (n-1) \, d \). Set the \( n \)-th term equal to 98 (the last term): \( 7 + (n-1) \, 7 = 98 \). Simplifying gives \( 7n = 98 \), so \( n = 14 \).

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

Use the sum formula for an arithmetic series: \( S_n = \frac{n}{2} (a + l) \). Substitute \( n = 14 \), \( a = 7 \), and \( l = 98 \) into the formula: \( S_{14} = \frac{14}{2} \, (7 + 98) \).

Step 4: Solve the Formula

Calculate the sum by evaluating the formula. \( S_{14} = 7 \, (105) = 735 \). Thus, the sum of the series is 735.

Key Concepts

sum of arithmetic series

The sum of an arithmetic series is the total when you add up all the terms in that series. This is a very useful concept, especially when dealing with large series. To find the sum of an arithmetic series, there is a straightforward formula you can use:

• First, determine the number of terms (\( n \)) in the series.
• Identify the first (\( a \)) and last terms (\( l \)) of the series.

The formula for the sum (\( S_n \)) is:\[ S_n = \frac{n}{2} (a + l) \]This formula takes the average of the first and last terms, multiplied by the number of terms. It simplifies the process of summing complex sequences with many terms. With this formula, you don’t need to add each term individually but can use basic known values for a quick calculation.

common difference

The common difference in an arithmetic series is the amount you add to each term to get the next term. It's a vital component of an arithmetic sequence because it defines how the series progresses.To find the common difference (\( d \)), choose any two consecutive terms in the series, then subtract the previous term from the next. For example, in the series 7, 14, 21, the common difference is:\[ d = 14 - 7 = 7 \]Understanding the common difference helps predict future terms in the series and is crucial for calculating other parameters like the number of terms or the sum of the series.

first term

The first term of an arithmetic series is the initial number in the sequence. It's denoted by (\( a \)) and is crucial because it serves as the starting point for calculating other elements of the series. In our example series, 7 is the first term.Knowing the first term lets you calculate further terms by using the common difference. It also plays a direct role in determining both the last term and the sum of the series. In practical applications, identifying the first term correctly is fundamental to setting the correct calculations in motion.

number of terms

Determining the number of terms (\( n \)) in an arithmetic series is essential to calculate the series' sum. Use the formula for the n-th term of an arithmetic series:\[ a_n = a + (n-1) \, d \]Here, \( a_n \) is the last term of the series, \( a \) is the first term, and \( d \) is the common difference.Solving for (\( n \)) involves setting the last term (\( a_n \)) equal to a known value in the series and solving. For example, if the last term is 98:\[ 7 + (n-1) \, 7 = 98 \]Simplify to find:\[ n = 14 \]The number of terms tells you how long the series is and directly impacts the calculations for the sum. Understanding this will help you determine how expansive your sequence is.