Problem 29 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{5}(2 i+1)$$

Step-by-Step Solution

Answer

The sum of the series is 35.

Step 1: Understanding the Series

The series given is \( \sum_{i=1}^{5}(2i + 1) \). This means we need to find the sum of the expression \( 2i + 1 \) as \( i \) takes on each integer value from 1 to 5.

Step 2: Evaluating Terms in the Series

Calculate each term by substituting the values 1 to 5 into \( 2i + 1 \):- When \( i = 1 \), \( 2(1) + 1 = 3 \)- When \( i = 2 \), \( 2(2) + 1 = 5 \)- When \( i = 3 \), \( 2(3) + 1 = 7 \)- When \( i = 4 \), \( 2(4) + 1 = 9 \)- When \( i = 5 \), \( 2(5) + 1 = 11 \)

Step 3: Summing the Series

Add the evaluated terms:\( 3 + 5 + 7 + 9 + 11 \).

Step 4: Calculating the Final Sum

Perform the addition:- \( 3 + 5 = 8 \)- \( 8 + 7 = 15 \)- \( 15 + 9 = 24 \)- \( 24 + 11 = 35 \)The sum of the series is 35.

Key Concepts

Sum of Series

An arithmetic series is essentially the sum of terms in an arithmetic sequence. In our original exercise, we are tasked with summing up a series given by the expression \( \sum_{i=1}^{5}(2i + 1) \). This means we are adding up values generated by the expression \( 2i + 1 \) as \( i \) increases from 1 to 5.
A series is not just a random collection of numbers; there is a logical structure to it. The sum is calculated by evaluating and adding each of these terms, giving us a single total value. This process helps us understand the behavior of the sequence over a range of values and allows us to find how much they all add up to. In mathematics, being able to sum such series is crucial in various fields, including calculus and financial mathematics.

Sigma Notation

Sigma notation is a concise way of writing a long sum of terms. It uses the Greek letter \( \Sigma \), which stands for "sum." In our example, \( \sum_{i=1}^{5}(2i + 1) \), the sigma notation tells us the following information:

• The lower limit, 1, is the starting point for the index \( i \).
• The upper limit, 5, is the ending point for the index \( i \).
• The expression \( 2i + 1 \) gives us the formula for each term in the series.

As i takes each integer value from 1 to 5, you substitute these numbers into the formula \( 2i + 1 \) to find each term. This concise notation is incredibly powerful because it allows for the easy representation of even complex series with numerous terms.

Evaluating Series Terms

Evaluating the terms of the series involves substituting the values of \( i \) into the expression \( 2i + 1 \). Consider this a step-by-step substitution process. For our example, you do it like this:

• When \( i = 1 \), you calculate \( 2(1) + 1 \), which equals 3.
• When \( i = 2 \), substitute to get \( 2(2) + 1 = 5 \).
• Continue this pattern up to \( i = 5 \), which results in a value of 11.

Evaluating the series terms is crucial because it gives you the necessary numbers to add. It also illustrates how each term relates to the index \( i \) and provides insights into how the series builds up through its sequence.

Step by Step Solution

To find the sum of the series \( \sum_{i=1}^{5}(2i + 1) \), you follow these steps:

• First, understand the series' format, noting that you need to sum terms generated by \( 2i + 1 \) for \( i \) values from 1 to 5.
• Evaluate the terms by sequentially substituting i's values from 1 to 5, calculating results: 3, 5, 7, 9, and 11.
• Add these calculated terms together. Begin with \( 3 + 5 = 8 \), then \( 8 + 7 = 15 \), \( 15 + 9 = 24 \), and finally \( 24 + 11 = 35 \).
• Therefore, the sum of the series is 35.

This step-by-step approach not only gives you the solution but helps reinforce the process of evaluating and summing terms of an arithmetic series. It paves the way for understanding more complicated series as you encounter them.