Problem 48 - 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 \(x_{5}=2\) $$\sum_{i=1}^{5}-x_{i}$$

Step-by-Step Solution

Answer

The sum evaluates to 0.

Step 1: Understand the Sum Notation

The notation \( \sum_{i=1}^{5} -x_{i} \) means that we need to calculate the sum of the expressions \(-x_{1}, -x_{2}, -x_{3}, -x_{4}, -x_{5}\). Each of these terms involves taking the negative of each \(x_i\).

Step 2: Calculate Each Negative Term

We need to calculate \(-x_{i}\) for each \(i\): - \(-x_{1} = -(-2) = 2\) - \(-x_{2} = -(-1) = 1\) - \(-x_{3} = -(0) = 0\) - \(-x_{4} = -(1) = -1\) - \(-x_{5} = -(2) = -2\)

Step 3: Evaluate the Sum

Now, sum up the calculated negative terms: \[ 2 + 1 + 0 + (-1) + (-2) \] Calculate sequentially: - \(2 + 1 = 3\) - \(3 + 0 = 3\) - \(3 + (-1) = 2\) - \(2 + (-2) = 0\)

Step 4: Verify and Conclude the Calculation

Verify that each step is done correctly. Summation of all terms results in \(0\). Thus, the final answer is 0.

Key Concepts

Summation

Summation is a mathematical operation that involves adding a sequence of numbers, known as addends or terms, together. In this exercise, we focus on evaluating the sum of the terms using sigma notation. Sigma notation, represented by the symbol \( \sum \), is a concise way of expressing the sum of a series of terms. It helps simplify expressions where repetitive addition is needed. You start by identifying each term you are going to add.

In the given exercise, the summation is represented as \( \sum_{i=1}^{5} -x_{i} \). This means we will take the sum over a sequence of expressions from \(i=1\) to \(i=5\). Each term is defined as \(-x_{i}\), meaning you must first convert each element \(x_i\) into its negative before summing them together.

Key points to remember about summation are:

• Simplifies lengthy addition problems
• Requires understanding of the limits of the sum (e.g., \(i=1\) to \(i=5\))
• Identifies each term to be added

Recognizing these components helps in efficiently calculating the summation.

Negative Numbers

Dealing with negative numbers is crucial in various mathematical calculations, including the evaluation of sums. Negative numbers are those less than zero, and mathematically, they are represented with a minus sign \((-x)\). When you negate a number that's already negative, it becomes positive. This is an important concept when evaluating sums that include negative terms.

In this exercise, each \(x_i\) term is converted into its negative. For example, when \(x_1 = -2\), negating \(-2\) gives you \(2\), because \(-(-2) = 2\). This same principle applies to each term where you flip their sign to become positive if they were negative, and negative if they were positive.

Key pointers when evaluating negative numbers:

• Negating a negative number results in a positive
• Ensure careful sign changes for accurate calculation
• Zero remains unaffected by negation

Evaluation

Evaluation refers to the process of determining the value of an expression. In terms of this exercise, once the negative values of \(x_i\) are computed for each \(i\), we evaluate the entire summation by adding these values together.

The procedure of evaluation follows these steps:

• Compute the individual negative values (e.g., \(-x_1 = 2\) when \(x_1 = -2\)
• Add all computed values in succession
• Always use orders of operations if needed

For our particular sequence: Start with \(2 + 1 = 3\), then add \(0\), followed by \(-1\), and finally \(-2\). Carefully compute each sum step-by-step to ensure that no calculation errors occur, resulting in the accurate total of \(0\). This exercise emphasizes the importance of orderly sequential calculation.

Mathematical Notation

Mathematical notation is a system of symbols used to write and communicate mathematical concepts precisely. In the context of summation, understanding mathematical notation like sigma provides clarity and structure to otherwise complex arithmetic tasks.

The use of \( \sum \) allows you to clearly define the operation (summation), the range of indices (\(i=1\) to \(i=5\)), and the specific expression being evaluated (\(-x_i\)). Each component has meaning and instructs you on how to proceed with evaluating the sum.

Pointers for understanding mathematical notation effectively:

• Know the symbols: Each symbol carries specific instructions
• Identify limits and operations clearly
• Symbols streamline writing and understanding of calculations

This notation is crucial for understanding and working efficiently with mathematical concepts, avoiding the confusion of lengthy written explanations.