Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=1}^{10} 3 n $$

Step-by-Step Solution

Answer

The sum of the series is 165.

Step 1: Understanding the Series

The problem provides the series in summation notation: \( \sum_{n=1}^{10} 3n \). This means we need to calculate the sum of terms of the form \(3n\) starting from \(n=1\) to \(n=10\).

Step 2: Writing the Series as a Sum of Terms

To express the series without using the summation symbol, we substitute values for \(n\) from 1 to 10: \(3\cdot1, 3\cdot2, 3\cdot3, \ldots, 3\cdot10\). Written out, the series is: \(3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30\).

Step 3: Calculating the Sum of the Series

Add up all the terms in the series: \(3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30\). We can perform the addition in pairs to simplify: \((3+30) + (6+27) + (9+24) + (12+21) + (15+18) = 33 + 33 + 33 + 33 + 33 = 165\).

Step 4: Verifying the Calculation with a Formula

For an arithmetic series where each term increases by a constant, we can use the formula \(S_n = \frac{n}{2}(a + l)\), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term. Here, \(n=10\), \(a=3\), and \(l=30\): \(S_{10} = \frac{10}{2}(3 + 30) = 5 \times 33 = 165\).

Key Concepts

Summation Notation

Summation notation is a concise way of representing long sums of numbers or expressions. It's especially useful when dealing with sequences and series. The notation is characterized by the Greek letter sigma (\( \Sigma \)), which is used to denote the sum of a sequence of terms:

• Above the sigma symbol, you'll find the number indicating the last term in the series.
• Below the sigma, the starting value of an index is indicated.
• To the right, an expression for the general term of the sequence is written.

In the example \( \sum_{n=1}^{10} 3n \), we are asked to calculate the sum of terms created by multiplying 3 with each integer from 1 to 10. This notation simplifies the representation of operations that would otherwise require writing out each term individually, making it much easier to work with complex calculations in mathematics.
Understanding how to read and interpret summation notation is essential for working with series and sequences efficiently.

Sum of Terms

To really understand a series, it's crucial to know how to break it down into its individual components. The idea of the "sum of terms" is about writing each term generated by the sequence in numerical form and adding them up.For example, using the series \( \sum_{n=1}^{10} 3n \), we plug in integers from 1 through 10 for \( n \) in the expression \( 3n \):

• \( n=1 \): \( 3 \times 1 = 3 \)
• \( n=2 \): \( 3 \times 2 = 6 \)
• \( n=3 \): \( 3 \times 3 = 9 \)
• \(...\)
• \( n=10 \): \( 3 \times 10 = 30 \)

Once all terms are expressed, the next step is to find their sum: \( 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30 \). For an easy addition process, grouping terms can help simplify calculations.The real-world application of breaking down a series into its individual terms is to manage large numbers of operations systematically, making it less intimidating and more approachable.

Arithmetic Sequence Formula

The arithmetic sequence formula is an invaluable tool for calculating sums quickly. An arithmetic sequence or series is one in which each term after the first is found by adding a constant difference to the previous term.If you know how many terms are in the series, the first term, and the last term, the sum \( S_n \) of the series can be efficiently calculated using the formula:\[ S_n = \frac{n}{2} (a + l) \]Where:

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

In the arithmetic series example \( \sum_{n=1}^{10} 3n \), we have 10 terms starting from 3 and ending at 30. By using the formula:\[ S_{10} = \frac{10}{2} (3 + 30) = 5 \times 33 = 165 \]Finding sums using this formula helps streamline complex calculations, ensuring accuracy and saving time when working with large datasets or calculations in academic and professional settings.