Problem 50 - 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 \(x_{5}=2\). $$\sum_{i=1}^{3}\left(x_{i}^{2}+1\right)$$

Step-by-Step Solution

Answer

The sum evaluates to 8.

Step 1: Identify the terms of the sum

The sum we need to evaluate is \( \sum_{i=1}^{3} \left(x_{i}^{2} + 1\right) \). This expression indicates that we will evaluate it for \( i = 1, 2, 3 \).

Step 2: Calculate the term for i = 1

Substitute \( x_1 = -2 \) into the expression. Calculate: \( (-2)^2 + 1 = 4 + 1 = 5 \).

Step 3: Calculate the term for i = 2

Substitute \( x_2 = -1 \) into the expression. Calculate: \( (-1)^2 + 1 = 1 + 1 = 2 \).

Step 4: Calculate the term for i = 3

Substitute \( x_3 = 0 \) into the expression. Calculate: \( 0^2 + 1 = 0 + 1 = 1 \).

Step 5: Sum the calculated terms

Add the results from steps 2, 3, and 4: \( 5 + 2 + 1 = 8 \). Thus, the evaluated sum is 8.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In this exercise, algebraic expressions like \( x_i^2 + 1 \) are being evaluated. Algebra often involves variables, such as \( x \), that can take on different values. Here, we use the index \( i \) to represent different positions in the sequence of expressions.

In our specific case, each \( x_i \) represents a term in a series, which allows us to apply algebraic rules like addition, multiplication, and exponentiation systematically.

When working with sums in algebra, it's essential to appreciate how these rules apply universally across different numbers, maintaining consistency and reliability. This is one of the central strengths of algebra, making it a powerful tool for solving a wide range of problems.

Evaluation of Expressions

The process of evaluating expressions involves systematically substituting values into an algebraic expression and simplifying. In our exercise, you evaluate the sum \( \sum_{i=1}^{3} (x_i^2 + 1) \) by substituting each value of \( x_i \) in sequence.

Here's how you do it:

• Substitution: Start by substituting each given \( x_i \) value into the expression. For example, when \( i = 1 \), use \( x_1 = -2 \).
• Simplification: Simplify the resulting expression by performing the arithmetic operations. For example, \( (-2)^2 + 1 = 4 + 1 = 5 \).
• Repetition: Repeat this process for each value of \( i \) in the range. This step-by-step approach ensures no errors are made and each term is accurately calculated.

Evaluation requires careful attention to detail, as small mistakes in calculations can lead to incorrect results. It’s about breaking down the problem into manageable parts and taking it one step at a time.

Steps to Solve Mathematical Problems

Solving mathematical problems often follows a structured approach. This helps in breaking down complex problems into simpler, manageable tasks. Here's a general roadmap that typically guides the problem-solving process:

• Understand the Problem: Clearly define what is being asked. Identify the terms and operations involved, as we did with the \( \sum \) notation.
• Plan the Approach: Decide on the sequence of operations. In our example, it's about substituting each value of \( x_i \) and then summing the results.
• Execute the Plan: Step through the calculations. Substitute each value and perform the required arithmetic as per the plan.
• Review the Results: Check the work for accuracy. Ensure that the summation of the terms was done correctly and revisit any steps if necessary.

This methodical approach ensures that you work through problems systematically, reducing the risk of oversight. It builds confidence in solving not only algebraic problems but also other mathematical challenges.