Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. 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! \\).

Step 1: Understanding Factorials

A factorial, denoted as \(n!\), is the product of all positive integers from \(1\) to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Step 2: Writing the Given Expression

The expression to evaluate is \( (n-1)! \cdot n \). This means we first calculate \( (n-1)! \) and then multiply by \( n \).

Step 3: Evaluating \\( (n-1) ! \\cdot n \\)

First, \( (n-1)! \) is \( (n-1) \times (n-2) \times \ldots \times 1 \). Therefore, \( (n-1)! \cdot n \) results in \( n \times (n-1) \times (n-2) \times \ldots \times 1 \).

Step 4: Comparing with \\(n!\\)

Knowing the definition of \( n! \), we have \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \). This matches exactly with the expression \( (n-1)! \cdot n \).

Step 5: Conclusion

The expression \( (n-1)! \cdot n \) is indeed always equal to \( n! \) for any positive integer \( n \) greater than 1. This results from the definitions and properties of factorials.

Key Concepts

Positive Integers

Positive integers are the set of all whole numbers greater than zero. This group excludes zero, negative numbers, and non-whole numbers like fractions or decimals. In mathematics, positive integers are often the building blocks for many concepts and principles. For instance:

• They start from 1 and go on to infinity: 1, 2, 3, 4, ...
• In factorial expressions, we only use positive integers.
• The set of positive integers is often denoted by the symbol \(\mathbb{N^+}\) or just \(\mathbb{N}\) without zero.

Understanding positive integers is fundamental when dealing with factorial-related problems as factorials are calculated starting from these numbers.

Factorial Properties

A factorial is represented by an exclamation mark (!). It means multiplying a series of descending positive integers. So, when you see \(n!\), it's shorthand for multiplying from \(n\) down to 1.

• For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
• Factorials grow rapidly; for instance, \(5! = 120\), which is significantly larger than \(4!\).
• An important property of factorials is that \((n) = n \cdot (n-1)!\), which simplifies their computation.

This recursive definition is why the expressions \((n-1)! \cdot n\) and \(n!\) are equivalent. This property shows us that multiplying \((n-1)!\) by \(n\) forms the complete multiplication needed for \(n!\). Understanding these properties helps not only in calculating factorials but also in solving many combinatorial problems.

Mathematical Expressions

Mathematical expressions involve numbers, variables, and operations acting together to form an equation or combination that represents a value. Factorials are a type of mathematical expression, often used to simplify complex problems or calculations, especially in permutations and combinations.

• They include operators such as addition (+), subtraction (-), multiplication (×), and division (÷).
• The equality sign (=) is often used to denote that two expressions are equivalent.
• Expressions can be simplified using properties like the associative, commutative, and distributive properties.

In factorial expressions, multiplication is the key operation, stringed together with a factorial property. Understanding mathematical expressions allows us to frame problems correctly and utilize factorial properties effectively for calculations, comparisons, and evaluations.