Problem 12 - A Graphical Approach to College Algebra

Question

If \(n\) is a positive integer greater than \(1,\) is \((n-1) ! \cdot n\) always equal to \(n ! ?\)

Step-by-Step Solution

Answer

Yes, \((n-1)! \cdot n\) is always equal to \(n!\) for \(n > 1\).

Step 1: Understand Factorial Notation

A factorial of a number, denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Step 2: Express Given Expression

We need to consider two expressions: \((n-1)! \cdot n\) and \(n!\). Write the first expression: \((n-1)!\) denotes the factorial of \(n-1\). So, \((n-1)! = (n-1) \times (n-2) \times \cdots \times 1\). Multiply this by \(n\) to get \((n-1)! \cdot n\).

Step 3: Compare with Factorial Definition

According to the definition of factorial, \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\). This is exactly the same as \((n-1)! \cdot n\).

Step 4: Conclude Equality

Since \((n-1)! \cdot n\) expands to the same multiplication as \(n!\), we conclude that \((n-1)! \cdot n = n!\) for any positive integer \(n > 1\).

Key Concepts

Understanding Integer

An integer is a whole number that can be positive, negative, or zero. In mathematics, when we talk about integers, we refer to numbers like \(-3, 0, 4\). These are numbers without fractions or decimals.
However, when solving factorial problems, we mainly focus on positive integers.
Here's why integers are essential in factorials:

• They help us understand the count of terms in a multiplication.
• Integers can be seen on the number line.

For example, in the factorial expression \(5!\), "5" is a positive integer that guides the product.

Exploring the Product

A product in mathematics is the result of multiplying numbers together.
For example, the product of \(3\) and \(4\) is \(12\) because \(3 \times 4 = 12\).
Products are crucial in understanding factorials:

• Factorials involve a series of products of consecutive integers.
• The operation builds from one integer to the next.

When calculating \(5!\), the product is \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
This operation shows how products form the core of factorial calculations.

Factorial Notation Demystified

Factorial notation, represented by an exclamation mark (!), is a way to convey a sequence of multiplied integers.
For example, \(n!\) means we multiply all positive integers from \(n\) down to \(1\).
Here's what makes factorial notation special:

• It simplifies expressing multiple multiplications.
• Commonly used in permutations and combinations.

When we write \(n!\), we're using a shorthand to show a complex multiplication.
In the equation we solved, \((n-1)! \cdot n = n!\), the notation helps clearly show the relationship between values.

Positive Integer Insights

Positive integers are the numbers we count with, starting at \(1\).
They are essential in defining factorials, as factorials only apply to non-negative numbers.
Here's why positive integers matter in our context:

• They ensure the operations in factorials make sense.
• The factorial of a positive integer \(n\) includes all numbers up to \(n\).

For example, in the expression \(n!\), \(n\) must be a positive integer like \(2, 3, 4,\) and so on.
This ensures that the factorial operation is valid and meaningful.