Problem 47 - 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 \(x_{5}=2\). $$\sum_{i=1}^{5}\left(2 x_{i}+3\right)$$

Step-by-Step Solution

Answer

The sum is 15.

Step 1: Understand the Question

We are asked to evaluate the sum \( \sum_{i=1}^{5}\left(2x_{i} + 3\right) \). This means we need to plug in each value of \( x_i \) into the expression \( 2x_i + 3 \), and then add them all together.

Step 2: Evaluate Each Term

Evaluate the expression for each \(x_i\):- For \(x_1 = -2\), the term is \(2(-2) + 3 = -4 + 3 = -1\).- For \(x_2 = -1\), the term is \(2(-1) + 3 = -2 + 3 = 1\).- For \(x_3 = 0\), the term is \(2(0) + 3 = 0 + 3 = 3\).- For \(x_4 = 1\), the term is \(2(1) + 3 = 2 + 3 = 5\).- For \(x_5 = 2\), the term is \(2(2) + 3 = 4 + 3 = 7\).

Step 3: Add the Terms

Sum the evaluated terms: \(-1 + 1 + 3 + 5 + 7 = 15\).

Step 4: Conclusion

The sum of the terms \( \sum_{i=1}^{5}\left(2x_{i} + 3\right) \) is 15.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols, such as letters, to represent numbers in equations and formulas. It provides a way to express general relationships and solve problems involving unknown values.
In the given exercise, we use an algebraic expression \(2x_i + 3\) to evaluate each term. An algebraic expression combines constants, variables, and arithmetic operations. Here:

• The constant is 3, which doesn't change its value.
• The variable is \(x_i\), which can take different values from the sequence \(-2, -1, 0, 1, 2\).
• The operation is multiplication followed by addition.

Understanding algebra is crucial because it forms the foundation for more complicated mathematics. It allows you to model real-world situations and enables problem-solving skills.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant difference to the previous term. This constant is called the common difference.
In this exercise, the sequence \(x_1, x_2, x_3, x_4, x_5\) is \(-2, -1, 0, 1, 2\). It is an arithmetic sequence with a common difference of 1, moving from one term to the next. This is because each term increases by 1 from the previous term.
Here's a closer look at the arithmetic sequence properties:

• The first term \(a = -2\).
• The common difference \(d = 1\).
• Each term \(x_n\) can be represented as \(a + (n-1)d\), helping in identifying terms quickly.

Recognizing sequences helps simplify problems involving series or patterns.

Expression Evaluation

Expression evaluation is the process of substituting values into an expression and executing the necessary arithmetic operations. It is a fundamental skill for solving mathematical problems.
In our problem, we're tasked with evaluating the sum \(\sum_{i=1}^{5}(2x_i + 3)\). Each \(x_i\) represents a value from the sequence \(-2, -1, 0, 1, 2\). For each \(x_i\):

• Start by plugging \(x_i\) into the expression \(2x_i + 3\).
• Perform the arithmetic operations: multiply \(x_i\) by 2, then add 3.

For example, when \(x_1 = -2\):

• Plug into the expression: \(2(-2) + 3\).
• Calculate: \(-4 + 3 = -1\).

After evaluating each term, add them up to get the total sum. This step-by-step process aids in ensuring accuracy and understanding the structure of expressions.