Problem 59 - Algebra 2

Question

State whether each statement is true or false when \(n=1\). Explain. $$ 1=\frac{n^{2}(n+1)^{2}}{4} $$

Step-by-Step Solution

Answer

True; both sides of the equation are equal when \(n=1\).

Step 1: Substitute n=1 into the formula

Substitute \(n=1\) into the given formula: \(\frac{n^{2}(n+1)^{2}}{4}\). This becomes \(\frac{1^{2}(1+1)^{2}}{4}\).

Step 2: Simplify the expression

Calculate \(1^{2}\), which equals 1, and \((1+1)^{2}\), which equals 4. Therefore, the expression becomes \(\frac{1 \times 4}{4}\).

Step 3: Perform the division

Divide the expression: \(\frac{1 \times 4}{4} = \frac{4}{4}\), which simplifies to 1.

Step 4: Compare with the left side

Compare the calculated value 1 to the left side of the equation: 1. Both sides are equal.

Step 5: Conclusion

Since both sides of the equation simplify to the same value, the statement is true for \(n=1\).

Key Concepts

Equation Solving

Solving algebraic equations involves finding the unknown variables that satisfy the given equation. In the scenario where you must state whether an equation is true or not for a specific value, such as when \(n = 1\), you simply input this value into the equation. This exercise asks us to determine if the simplified form equals the other side of the equation.

• The "unknown" in the equation is \(n\), and we follow the given instructions to test if the statement holds.
• The process requires verifying through substitution and simplification to ensure both sides remain equal.
• If both sides equal the same value, then the statement is true for that particular value.

By practicing this method, you build the foundational skills needed for more complex equations. Remember, the key is precise calculation and verification.

Substitution Method

The substitution method is a straightforward way to simplify complex expressions by replacing variables with specific values. It's like swapping puzzle pieces to see if they fit. In this exercise, you substitute \(n=1\) into the equation \(\frac{n^2(n+1)^2}{4}\). Here’s how it works:

• Replace \(n\) with \(1\) everywhere it appears in the equation.
• The formula changes to \(\frac{1^2(1+1)^2}{4}\), simplifying the components within parentheses first.

Substitution helps in transforming the equation into a numeric calculation, making it easier to compare the results with the other side of the equation. By practicing substitution, you enhance your ability to tackle more intricate problems in algebra.

Simplification Steps

Simplification involves reducing an expression to its simplest form. After substitution, the simplifying steps entail calculating powers, products, and any operations as in basic arithmetic operations. In this exercise, you perform simplification after substituting \(n = 1\).

• Calculate \(1^2\), which is 1.
• Then, \((1+1)^2\) results in 4.
• This turns the formula into \(\frac{1 \times 4}{4}\).
• Finally, simplify \(\frac{4}{4}\), which results in 1.

The simplification ensures that the equation is as straightforward as possible, making it easy to compare both sides. It's a vital skill in algebra, allowing for clarity and correctness in solutions.