Problem 99 - College Algebra

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{-1}^{2}(-1) / 2-0$$

Step-by-Step Solution

Answer

The original statement is False. The correct statement should be \(\sum_{i=0}^{2}(-1)^i / 2-0\) and the value of the corrected expression is \(1/2\)

Step 1: Understanding the Summation Notation

For a typical notation, where \(i\) is the index of summation, \(a\) and \(b\) are the lower and upper limits of the summation: \( \sum_{i=a}^{b} f(i) \), \(i\) ranges over all integers from \(a\) to \(b\) inclusive. Here the indices seem to be negative and positives, which could be a mistake.

Step 2: Correcting the Summation Index

The summation in the stated problem is given from \(-1\) to \(2\) which is unusual. Usually, summation starts from a non-negative integer. Therefore, we change the lower limit to \(0\), making the corrected statement as \( \sum_{i=0}^{2}(-1)^i / 2-0 \)

Step 3: Evaluate the Corrected Expression

Substitute \(i\) into the expression and sum the results, we get: \((-1)^0/2-0 + (-1)^1/2-0 + (-1)^2/2 - 0 = 1/2 + -1/2 + 1/2 = 1/2\)

Key Concepts

Algebra Problem Solving

Problem-solving in algebra involves understanding the question, identifying relevant mathematical principles, and applying these principles to find a solution. To tackle algebra problems effectively, it's important to break them down into smaller, manageable steps.

When faced with a summation problem, like in the exercise provided, the first step is to clearly comprehend the summation notation, which is a compact way to represent the addition of a series of numbers generated by a function of varying integers. Once you grasp the notation, you can then address any issues in the expression, such as incorrect limits of summation.

In the given exercise, the student must correct the summation indices, ensuring they align with conventional practices. After making these corrections, the next step involves computing the sum by evaluating the expression at each integer between the corrected limits, thus solving the problem.

For successful algebra problem solving, remember to:

• Understand the given problem and notation.
• Identify any errors in the indices or expressions.
• Apply the corrected principles to evaluate the sum.

College Algebra

College algebra serves as the foundation for advanced studies in various scientific and technical fields. Students encounter a variety of abstract problems that require the manipulation of mathematical expressions and the understanding of functions, matrices, and complex numbers.

The exercise on summation notation is a typical example of the kind of abstract reasoning required in college algebra. To excel, one must become adept at interpreting symbolic expressions and executing operations as per algebraic rules.

The ability to correct a potentially unusual index in a summation, as with the given problem, is a critical skill. It underscores the importance of paying close attention to the details in mathematical expressions. Students should practice solving a variety of problems to build a deep understanding of algebraic concepts.

Key elements for mastering college algebra include:

• Mastering fundamental algebraic operations and functions.
• Developing problem-solving strategies.
• Attention to detail in mathematical expressions and conventions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that together represent a specific value or set of values. In the context of summation, an expression defines the terms being summed.

Understanding the syntax and conventions of expressions, such as the use of parentheses and the significance of indices, is crucial. Summation notation, for example, compactly expresses the addition of terms generated by substituting a range of integers into the given function or expression.

In the step by step solution provided, the mathematical expression within the summation notation required modification to reflect a more conventional approach, starting the index at zero. The final step involves substituting the appropriate integers into the corrected expression and performing the summation to arrive at the correct result.

To comprehend and manipulate mathematical expressions effectively, keep in mind to:

• Learn and adhere to mathematical conventions and notation.
• Understand the structure and components of expressions.
• Practice evaluating expressions accurately by substituting in values where necessary.