Question

True or False If \(n \geq 2\) is an integer, then $$ n !=n(n-1) \cdots 3 \cdot 2 \cdot 1 $$

Short answer

True

Step-by-step solution

- Understand the Factorial Notation

Factorial of an integer \(n\) is defined as the product of all positive integers less than or equal to \(n\). It is denoted by \(n!\).

- Write the Definition of Factorial

For an integer \(n \geq 2\), the factorial \(n!\) is given by: \[ n! = n \times (n-1) \times (n-2) \times \text{...} \times 3 \times 2 \times 1 \]

- Compare with the Given Expression

Compare the definition of the factorial with the given expression: The given expression is \( n(n-1) \text{...} 3 \times 2 \times 1 \). This matches exactly with the definition of \(n!\) for \(n \geq 2\).

- Conclude the Statement

Since the given expression matches the definition of factorial, the statement is True.

Key concepts

Factorial Definition

Factorials are a fundamental concept in mathematics, especially in combinatorics and calculus. Simply put:
The factorial of a number helps count the ways objects can be arranged.
For any positive integer, the factorial is defined as the product of all integers from 1 up to that number.
Let's break it down:

• If we denote a positive integer with 'n', its factorial is written as 'n!'.
• This symbol '!' is unique to the factorial and shouldn't be confused with basic punctuation.
• For example, if n=4, then 4! = 4 × 3 × 2 × 1 = 24.

Generally, for any integer n (assuming n ≥ 2):

n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1. This means multiplying n by every positive integer less than it, right down to 1.
Underling factorials help in simplifying numerous mathematical problems and finding solutions efficiently.

Integer Properties

Integers are whole numbers that can be either positive, negative, or zero. Integer properties are essential when dealing with factorials because factorials are specifically defined for non-negative integers. A few notable properties of integers include:

• Closure: The set of integers is closed under addition, subtraction, and multiplication. That means adding, subtracting, or multiplying any two integers will always result in another integer. However, division is an exception unless the divisor is a factor of the dividend.
• Associative Property: For any integers a, b, and c, both (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c) hold true. This property allows us to re-group integers in any manner for addition and multiplication.
• Commutative Property: For any integers a and b, a + b = b + a and a × b = b × a. This means the order in which we add or multiply integers doesn't affect the outcome.
• Distributive Property: For any integers a, b, and c, a × (b + c) = (a × b) + (a × c). This property connects addition and multiplication operations.

When applying integers to factorials, remember:

• Factorials are defined only for non-negative integers (0, 1, 2, 3, ...).
• The factorial of a negative integer or a decimal does not exist within the standard definition.
• The smallest factorial is 0!, which is defined as 1. By definition, 1! is also 1.

Understanding integer properties helps in manipulating expressions and solving problems involving factorials more effectively.

Mathematical Proof

Mathematical proofs provide a logical way to establish the truth of a statement using axioms, definitions, and previously established results. Let's consider the proof that validates the given statement in the original exercise.

Step 1: Understand the given statement. - We're asked if the expression n! = n(n-1)...3×2×1 is true for any integer n ≥ 2.

• Step 2: Use the definition of factorial. - By definition, the factorial of an integer 'n', denoted as n!, is the product of all positive integers less than or equal to n.

• Step 3: Compare with given expression. - The expression provided is exactly the same as the factorial definition: n(n-1)...3×2×1.

• Step 4: Conclude the statement.
- Since the definition and the given expression match perfectly, we can conclude that the statement is true.

Insights into proofs:
Proofs use a sequence of logical steps to ensure every assertion is definitively shown or supported. This step-by-step approach minimizes errors and provides a clear path from assumptions to conclusions. When working with proofs:

• Always start by defining or understanding the terms and conditions given in the statement.
• Break the problem into smaller logical parts, tackling each step-by-step.
• Ensure every step follows logically from the previous one, using definitions, properties, or previously established results.

Proofs are crucial for advancing mathematical concepts and help affirm knowledge.