Problem 22 - Algebra 2

Question

For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. $$ \sum_{n=1}^{5}(0.2 n-0.2) $$

Step-by-Step Solution

Answer

The number of terms is 5. The first term is 0 and the last term is 0.8. The sum of the series is 2.

Step 1: Find the number of terms

The number of terms is determined by the range of \(n\) in the summation. Here, \(n\) ranges from \(1\) to \(5\), making a total of \(5\) terms.

Step 2: Find the first term

To find the first term, replace \(n\) in the equation with \(1\). So, the equation becomes \(0.2*1 - 0.2 = 0\)

Step 3: Find the last term

To find the last term, replace \(n\) in the equation with \(5\). So, the equation becomes \(0.2*5 - 0.2 = 0.8\)

Step 4: Evaluate the series

The value of the series is the sum of all its terms. Using the formula for each term: \(0.2n - 0.2\), calculate each term for \(n\) from \(1\) to \(5\) and add them together. So the value of the series becomes: \[ (0.2*1 - 0.2) + (0.2*2 - 0.2) + (0.2*3 - 0.2) + (0.2*4 - 0.2) + (0.2*5 - 0.2) = 0 + 0.2 + 0.4 + 0.6 + 0.8 = 2 \]

Key Concepts

Number of Terms

The number of terms in an arithmetic series is essential to determine, as it tells us how many numbers are involved in the series before we sum them up. In the original exercise, the summation notation \( \sum_{n=1}^{5} (0.2n - 0.2) \) indicates that \( n \) ranges from \( 1 \) to \( 5 \). This range means that we have a total of 5 terms.

To identify the number of terms:

• Look at the lower and upper bounds in the summation notation.
• Subtract the lower bound from the upper bound.
• Add one to the result to include both endpoints.

In this case, it's straightforward because the difference is exactly 4, and adding 1 results in 5 terms. Understanding this aspect helps you structure the series correctly.

First Term

The first term of a series is found by substituting the smallest value of \( n \) into the expression inside the summation. Here, for \( \sum_{n=1}^{5} (0.2n - 0.2) \), you substitute \( n = 1 \) into the equation \( 0.2n - 0.2 \).

For the first term:

• Replace \( n \) with \( 1 \) in the expression.
• Calculate the result to find the first term's value.

So, substituting gets us \( 0.2 \times 1 - 0.2 = 0 \). This process determines the initial value from which the series begins. It's a simple substitution task that sets the stage for understanding the transition of the series.

Last Term

The last term in an arithmetic series is discovered by simply using the largest value of \( n \) in your range. For the expression \( \sum_{n=1}^{5} (0.2n - 0.2) \), you will substitute \( n = 5 \) into the equation.

To find the last term:

• Substitute the maximum value of \( n \) into the expression.
• Perform the calculation to determine the value of the last term.

Doing this, we find \( 0.2 \times 5 - 0.2 = 0.8 \). Knowing the last term is crucial because it tells you the final value the series reaches. It completes your understanding of the series' range and progression.

Evaluating Series

Evaluating an arithmetic series is the process of calculating the sum of all the terms within the specified range. In this exercise, the expression \( 0.2n - 0.2 \) gives us the value of each term, and we need to add all terms from \( n = 1 \) to \( n = 5 \).

Steps to evaluate the series include:

• Calculate each term individually by substituting values of \( n \).
• Add these individual results together.

For this example, the process is:

• \( 0.2 \times 1 - 0.2 = 0 \)
• \( 0.2 \times 2 - 0.2 = 0.2 \)
• \( 0.2 \times 3 - 0.2 = 0.4 \)
• \( 0.2 \times 4 - 0.2 = 0.6 \)
• \( 0.2 \times 5 - 0.2 = 0.8 \)

Adding these together, we achieve the sum of the series: \( 0 + 0.2 + 0.4 + 0.6 + 0.8 = 2 \). Evaluating the series gives you the total sum, which is the final goal of the process.