Problem 21 - Algebra 2 and Trigonometry

Question

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{n=0}^{9}(20-2 n) $$

Step-by-Step Solution

Answer

The sum of the arithmetic series is 110.

Step 1: Determine the Terms of the Series

We start by identifying the terms of the series. The general term given is \(20-2n\), and the series ranges from \(n=0\) to \(n=9\). Thus, we'll substitute each value of \(n\) from 0 to 9 into the expression \(20-2n\) to find each term.- For \(n=0\), the term is \(20-2(0) = 20\).- For \(n=1\), the term is \(20-2(1) = 18\).- For \(n=2\), the term is \(20-2(2) = 16\).- For \(n=3\), the term is \(20-2(3) = 14\).- For \(n=4\), the term is \(20-2(4) = 12\).- For \(n=5\), the term is \(20-2(5) = 10\).- For \(n=6\), the term is \(20-2(6) = 8\).- For \(n=7\), the term is \(20-2(7) = 6\).- For \(n=8\), the term is \(20-2(8) = 4\).- For \(n=9\), the term is \(20-2(9) = 2\).

Step 2: Write the Series as a Sum of Terms

Now, we'll write the series as a sum of the terms calculated in Step 1:\[20 + 18 + 16 + 14 + 12 + 10 + 8 + 6 + 4 + 2\]

Step 3: Sum the Series

Next, we calculate the sum of the series:\(20 + 18 + 16 + 14 + 12 + 10 + 8 + 6 + 4 + 2 = 110\)

Key Concepts

Sum of Terms

To understand the sum of terms in an arithmetic series, we first need to know what terms are. In an arithmetic sequence, each number is called a term. These terms are created by adding or subtracting a fixed number, known as the common difference, to the previous term.
In the exercise provided, we have an expression for the general term: \(20-2n\). Starting from \(n=0\) to \(n=9\), each term is calculated by substituting the value of \(n\) into the expression.
This calculation generates the following sequence of terms: 20, 18, 16, 14, 12, 10, 8, 6, 4, and 2. This is simply a list of numbers where each subsequent number is obtained by subtracting 2 from the previous number, demonstrating the arithmetic nature of this series.

Arithmetic Series Calculation

An arithmetic series is essentially the sum of a sequence of numbers where each term increases or decreases by a constant value. Calculating the series means adding up all the individual terms derived from your expression.
For the given problem, our task was to calculate \(20+18+16+14+12+10+8+6+4+2\).
Performing this addition involves cascading through the list of numbers and step-by-step summing each pair until no numbers are left. This can start with \(20 + 18 = 38\), then add the next term, 16, resulting in \(38 + 16 = 54\), and continue this process.

• Series starts at 20 and ends at 2
• Common difference is -2
• Arithmetic operation is simple addition performed sequentially

By summing all these values as described, we find our final sum of the series to be 110.

Series Summation Steps

The process of series summation can be methodical. It's about organizing the addition of terms step by step. When working with arithmetic series, especially like the one in this exercise, it's crucial to keep these steps clear and efficient.

To easily follow and sum an arithmetic series, you should:

• Identify the formula for the general term, in this case, \(20 - 2n\).
• Calculate each term value by substituting \(n\) from 0 through 9.
• Write down the sequence of terms: 20, 18, 16, 14, 12, 10, 8, 6, 4, 2.
• Add these terms sequentially to find the total sum.

By maintaining this order, you simplify the process of handling longer series and ensure accuracy, arriving at the series' sum, which here is 110.
Practicing and becoming comfortable with these summation steps will significantly aid in solving any arithmetic series problem efficiently.