Problem 45 - A Graphical Approach to College Algebra

Question

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\). $$\sum_{i=1}^{5} x_{i}$$

Step-by-Step Solution

Answer

The sum is 0.

Step 1: Understand the Sum Notation

The expression \( \sum_{i=1}^{5} x_{i} \) represents the sum of the sequence \( x_i \) from \( i = 1 \) to \( i = 5 \). This means we are to add each of the values of \( x_i \) from \( x_1 \) to \( x_5 \).

Step 2: Identify the Values

We are given the values \( x_1 = -2, x_2 = -1, x_3 = 0, x_4 = 1, \) and \( x_5 = 2 \). These are the terms that we need to sum together.

Step 3: Perform the Addition

Calculate the sum by adding the individual terms: \( x_1 + x_2 + x_3 + x_4 + x_5 = -2 + (-1) + 0 + 1 + 2 \).

Step 4: Calculate the Result

First, add \( -2 + (-1) = -3 \). Next, add \( -3 + 0 = -3 \). Then add \( -3 + 1 = -2 \). Finally, add \( -2 + 2 = 0 \).

Step 5: Confirm the Result

Verify the addition is correct. Start with \( -2 + (-1) = -3 \), then \( -3 + 0 = -3 \), followed by \( -3 + 1 = -2 \), and concluding with \( -2 + 2 = 0 \). The calculations are confirmed correct.

Key Concepts

Sequence

A sequence is simply an ordered list of numbers. Each number in the sequence is called a term. In our exercise, we have a sequence with five terms: \( x_1 = -2, x_2 = -1, x_3 = 0, x_4 = 1, \) and \( x_5 = 2 \). This list is important because each term has a specific position.

The subscript in \( x_i \) indicates the position of the term in the sequence. For instance, \( x_3 \) is the third term. Understanding the order is key to applying operations like summation correctly.

• Terms are components of a sequence.
• Order is crucial in a sequence.
• Each term can be used in arithmetic operations.

This concept of sequence lays the foundation for many mathematical topics, as it helps us understand sets of numbers and their behaviors.

Sum Notation

Sum notation is a shorthand way of writing the addition of multiple elements in a sequence. It uses the Greek letter \( \Sigma \) to represent the sum. For example, \( \sum_{i=1}^{5} x_i \) tells us to add the terms from \( x_1 \) to \( x_5 \).

This notation is efficient because it avoids writing long sums repeatedly. Here's a breakdown of the notation:

• \( \Sigma \) symbolizes summation.
• The index \( i=1 \) tells us where to start.
• \( 5 \) indicates the last term to include.
• \( x_i \) shows the sequence whose terms we add.

Understanding sum notation is essential for simplifying complex arithmetic and algebraic expressions, especially when working with long sequences.

Addition

Addition is one of the most basic arithmetic operations and is simply the process of bringing two or more numbers together to make a new total. In our exercise, we need to add the terms of the sequence \( -2, -1, 0, 1, 2 \).

The sum of these numbers is calculated as follows:

• First, add \(-2 + (-1) = -3\).
• Next, continue with \(-3 + 0 = -3\).
• Then, add \(-3 + 1 = -2\).
• Finally, \(-2 + 2 = 0\).

The result of these additions is \( 0 \). This process of step-by-step addition ensures accuracy and helps verify results, demonstrating that addition is cumulative and associative, which is integral to solving more complex problems.