Problem 47 - Discrete Mathematics with Applications

Question

Verify each. $$(n+1) !-n !=n(n !)$$

Step-by-Step Solution

Answer

To verify the given equation \((n+1)! - n! = n(n!)\), we expand the factorials, substitute them back into the equation, and factor out the common term \(n!\). This results in \(n! \cdot n = n(n!)\), which confirms that the equation holds true.

Step 1: Expand the factorials

First, let's expand the factorials on the left-hand side of the equation: \((n+1)! = (n+1) \cdot n \cdot (n-1) \cdot ... \cdot 2 \cdot 1\) \(n! = n \cdot (n-1) \cdot ... \cdot 2 \cdot 1\)

Step 2: Substitute expanded factorials into the equation

Now, substitute the expanded factorials back into the equation: \((n+1) \cdot n \cdot (n-1) \cdot ... \cdot 2 \cdot 1 - n \cdot (n-1) \cdot ... \cdot 2 \cdot 1 = n(n!)\)

Step 3: Factor out the common term

Notice that both terms on the left-hand side have a common term, \(n \cdot (n-1) \cdot ... \cdot 2 \cdot 1\), which is equivalent to \(n!\). Let's factor this out: \(n! \left[(n+1) - 1 \right] = n(n!)\)

Step 4: Simplify the expression

Now, simplify the expression inside the brackets: \(n! \cdot n = n(n!)\)

Step 5: Verify the equation

As we can see, the left-hand side and the right-hand side of the equation are equal: \((n+1)! - n! = n(n!)\) This verifies the given equation.

Key Concepts

Factorials: Understanding the "!" Notation

Factorials are a crucial concept in mathematics that help us count and arrange items. The notation for a factorial is an exclamation mark, like this: \( n! \). It represents the product of an integer and all the integers below it down to 1. For example, \( 5! \) (read as "five factorial") equals \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

When you see the factorial notation, it means you take the number and multiply it by all positive whole numbers less than it. Factorials grow very quickly, and they are especially useful in permutations and combinations.

Here are a few key facts about factorials:

• The factorial of 0, \( 0! \), is defined to be 1. This is a special rule that helps with mathematical consistency.
• Factorials are only defined for non-negative integers (0, 1, 2, 3, ...).

Understanding factorials makes it easier to handle problems where you need to arrange items in different ways, like seating people around a table or creating passwords with specific requirements.

Mathematical Proof: Steps to Confirm a Statement

A mathematical proof is like a logical argument that demonstrates why a mathematical statement is true. It involves a series of steps that start from known facts and use logical reasoning to arrive at the statement we want to prove.

In our exercise, we want to prove the equation \((n+1)! - n! = n(n!)\). To do this, we start by expanding the factorials. This means we write each factorial in its multiplied form. Next, substitute the expanded forms into the equation.

The process continues by looking for common factors, which helps in simplifying the equation. Then, we factor out these common terms and perform algebraic manipulations. Finally, we simplify the expression to reach the desired conclusion that both sides of the equation are indeed equivalent.

By laying out each step clearly and logically, proofs help us understand complex statements and verify their validity. Proofs are fundamental to mathematics as they ensure that the results we use and build upon are universally true.

Equation Verification: Ensuring Both Sides Match

Verifying an equation means showing that both sides of the equation are equal, confirming its correctness. Equation verification involves working through mathematical operations to check if the left-hand side equals the right-hand side after simplification.

For the equation \((n+1)! - n! = n(n!)\), we start by breaking down each term. After expanding and simplifying, it's easier to inspect for any discrepancies or mistakes in our arithmetic.

The key steps in verifying include:

• Careful substitution by replacing terms with their expanded forms.
• Identifying and factoring out common terms to simplify expressions.
• Simplifying both sides to show they are indeed equal.

Equation verification is essential, especially in complex equations, as it reassures us that the operations and substitutions are carried out accurately. It's a final check to ensure the mathematical truth of a statement.