Problem 99 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The statement is false. The corrected statement should be \( \sum_{i=1}^{2}(-1)^{i} 2^{i}=2 \).

Step 1: Understanding the Variables

We can notice that there is a typo in the question. The variable 'j' is not clearly defined. However, we can assume the series meant to use 'i' instead of 'j', for there is no 'j' in the problem. It should be: \( \sum_{i=1}^{2}(-1)^{i} 2^{i} \).

Step 2: Substituting the Variables

Next, we need to substitute the variable 'i' with its corresponding values 1 and 2. The sum becomes: \( (-1)^{1} 2^{1} + (-1)^{2} 2^{2} \).

Step 3: Simplifying the Expression

Calculating the individual expressions and adding them, we get: \( -2 + 4 = 2 \). The sum does not equal 0.

Step 4: Making Corrections to the Statement

Since the result we got is 2 and not 0 as the statement indicated, the statement is false. To correct the statement, it should be changed to \( \sum_{i=1}^{2}(-1)^{i} 2^{i} = 2 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They're foundational in mathematics as they allow us to generalize numbers and arithmetic operations. In this context, the expression \( (-1)^i 2^i \) is used within summation notation. Here, 'i' is a variable that takes integer values. Each value of 'i' substitutes into the expression to evaluate a part of the summation. This makes the concepts flexible and powerful in handling patterns and sequences.

• Variables like 'i' are placeholders representing numbers.
• Operations include both arithmetic and exponentiation, such as \(( -1)^i\) and \(2^i\).

Understanding how to manipulate and evaluate algebraic expressions is critical. It involves knowing how to apply arithmetic operations and knowing the properties of exponents. This knowledge makes it possible to simplify or solve various mathematical problems.

Series Evaluation

Series evaluation involves calculating the sum of a sequence of terms. In our exercise, this is expressed using summation notation:\( \sum_{i=1}^{2}(-1)^{i} 2^{i} \).This notation is a compact way to represent the sum of multiple terms generated by changing the variable 'i' within a specified range.
To evaluate, you:

• Identify the range for 'i', which dictates how many terms to evaluate.
• Substitute each integer value of 'i' into the expression.
• Compute each term's value and then add these values to find the total.

For instance:

• When \(i = 1\): the term is \((-1)^1 (2)^1 = -2\)
• When \(i = 2\): the term is \((-1)^2 (2)^2 = 4\)

Adding these gives \(-2 + 4 = 2\). Understanding how to evaluate such expressions helps in various analytical contexts like probability, calculus, and finite mathematics.

Correcting Mathematical Statements

Correcting mathematical statements is an important skill. It involves identifying mistakes and knowing how to fix them to ensure accuracy in results. In the exercise, an error was found in the variable used in the expression. Initially, the statement had 'j' instead of 'i', which was not defined. This was a typographical error, leading to incorrect algebraic evaluation.
To correct this:

• First, discern if a variable is improperly used through misalignment with the problem context or setup.
• Ensure all terms and variables are properly defined and understood in your work to avoid such mistakes.
• Recalculate with the correct expression to reach the right result.

In our example, the necessary correction involved replacing 'j' with 'i'. After correctly resolving, the statement was shown to differ from zero, correcting to \( \sum_{i=1}^{2}(-1)^{i} 2^{i} = 2\).This highlights the importance of meticulous attention to detail and verifying each term's validity when working with mathematical expressions.