Problem 58 - Algebra 2

Question

State whether each statement is true or false when \(n=1\). Explain. $$ 1=\frac{(n+1)(2 n+1)}{2} $$

Step-by-Step Solution

Answer

False, because substituting \(n=1\) results in an unequal equation.

Step 1: Substitute Given Value of n

First, we substitute the given value of \(n = 1\) into the right-hand side of the equation. This gives:\[\frac{(1+1)(2 \times 1+1)}{2} = \frac{(2)(3)}{2}\]

Step 2: Simplify the Expression

Next, we simplify the expression:\[\frac{(2)(3)}{2} = \frac{6}{2} = 3\]

Step 3: Compare the Two Sides

Now, compare the left-hand side, which is \(1\), with the simplified right-hand side, \(3\). Since \(1 eq 3\), the equation \(1=\frac{(n+1)(2 n+1)}{2}\) does not hold true when \(n=1\).

Step 4: Conclusion: Determine the Truth Value

Since the left-hand side \(1\) is not equal to the right-hand side \(3\), the statement is false when \(n=1\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra and is a crucial first step in evaluating expressions. It involves replacing variables with given values to simplify the work that's needed afterwards. Often, exercises will provide specific values for variables - like "when \( n = 1 \)" in this exercise - to help you practice substitution.

• Find the variable in the equation (e.g., \( n \)).
• Replace it with the given value throughout the equation to see how it changes.

For instance, substituting \( n = 1 \) in \( \frac{(n+1)(2n+1)}{2} \) requires you to substitute \( n \) wherever it appears, yielding \( \frac{(1+1)(2\times1+1)}{2} \), simplifying the scenario considerably.

Simplification

Simplification is the process of reducing an expression into its simplest form. This makes it easier to interpret and compare with other expressions or values. After substitution, simplification follows to reduce calculations.

• Group like terms, perform any arithmetic operations, and simplify fractions if possible.
• Maintain clarity by working step-by-step so that there’s less room for error.

In our example, after replacing \( n \) with \( 1 \), the expression became \( \frac{(2)(3)}{2} \). Simplifying this gives \( \frac{6}{2} = 3 \). It's crucial to simplify correctly to compare values at the next stage.

Equation Solving

Equation solving involves working with both sides of an equation to find unknown values or verify statements. Often, it requires balancing both sides to ensure equality. Here, you're not solving for a variable, but instead confirming if a statement is correct under given conditions.

• Substitute the known values and simplify.
• Compare the left and right-hand sides of the equation after simplification.

In the provided scenario, after substitutions and simplifications, compare the left side \(1\) with the right side result \(3\). Noting that they are unequal means the statement doesn’t hold under the given conditions.

Truth Values

Truth values help you make claims about whether statements are true or false. In algebra, after completing substitution and simplification, examining the truth value of an expression helps validate results.

• When both sides of an equation are equal, the statement is true.
• If they differ, as they do in this exercise (\(1eq3\)), the statement is false for the tested value.

Determining truth values is essential for validating whether a mathematical claim holds true, such as when setting conditions or proving statements in algebraic contexts. Here, seeing that \(1 eq 3 \) shows that the condition does not satisfy the original equation, confirming its falsity."