Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{4}\left(4-6 x_{i}\right)$$

Step-by-Step Solution

Answer

The sum of the terms is 28.

Step 1: Calculate First Term

Determine the first term of the sum, where \(i=1\). Substitute \(x_1 = -2\) into the expression \(4 - 6x_i\). \[ 4 - 6(-2) = 4 + 12 = 16 \]

Step 2: Calculate Second Term

Determine the second term of the sum, where \(i=2\). Substitute \(x_2 = -1\) into the expression \(4 - 6x_i\). \[ 4 - 6(-1) = 4 + 6 = 10 \]

Step 3: Calculate Third Term

Determine the third term of the sum, where \(i=3\). Substitute \(x_3 = 0\) into the expression \(4 - 6x_i\). \[ 4 - 6(0) = 4 \]

Step 4: Calculate Fourth Term

Determine the fourth term of the sum, where \(i=4\). Substitute \(x_4 = 1\) into the expression \(4 - 6x_i\). \[ 4 - 6(1) = 4 - 6 = -2 \]

Step 5: Sum All Terms

Add all the calculated terms together: \(16 + 10 + 4 - 2\). \[ 16 + 10 + 4 - 2 = 28 \]

Key Concepts

Series Evaluation

Evaluating a series involves calculating the sum of a sequence of numbers. In this exercise, the series to evaluate is the expression \( \sum_{i=1}^{4}(4-6x_i) \). Here, the goal is to find the sum by calculating each term separately and then summing them up.

When evaluating a series, the challenge often lies in correctly substituting each value into the expression and performing the operations systematically.

• First, identify each term in the series.
• Next, substitute the values provided – in this case, \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1\) – into the expression.
• Finally, sum the results to find the total sum.

This process provides insight into how each element contributes to the final result. Understanding the method of series evaluation is crucial for more complex mathematical concepts, as it forms the basis for many areas of mathematics.

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics and essential for solving problems like the one given. These operations typically include addition, subtraction, multiplication, and division. In this exercise, primarily addition and subtraction are used, alongside multiplication when substituting values.

Let's consider the expression \(4 - 6x_i\) used in the series.

• When a value of \(x_i\) is substituted, the term \(6x_i\) involves multiplication.
• This multiplication results in either a negative or positive value depending on the sign of \(x_i\).
• Then, subtract the result from 4, where the order of subtraction is dependent on the resulting sign.

These operations are performed for each term. Mastering arithmetic operators is critical; they allow one to manipulate and simplify expressions effectively. In complex calculations, they remain pivotal in breaking down and solving problems.

Substitution Method

The substitution method involves replacing variables with specific numbers to simplify an expression or solve an equation. In this exercise, substitution is used to evaluate the series by determining the precise value of each term.

This exercise illustrates the mechanic of substitution:

• Identify each variable, \(x_i\), that needs to be replaced.
• Substitute given values, like \(x_1 = -2\), into the expression \(4 - 6x_i\).
• Calculate the value of the expression with the substituted number.

By understanding substitution, students can easily navigate through formulas and equations. This method not only simplifies calculations but also forms a foundation for solving system of equations and integrating functions in advanced mathematics.