Problem 31 - A Graphical Approach to College Algebra

Question

Prove each statement by mathematical induction. $$\text { If } n \geq 4, \text { then } n !>2^{n}$$

Step-by-Step Solution

Answer

The statement \( n! > 2^n \) holds for all \( n \geq 4 \).

Step 1: Base Case Verification

Begin by checking the base case where \( n = 4 \). Compute \( 4! \) and \( 2^4 \). We have \( 4! = 24 \) and \( 2^4 = 16 \). Since \( 24 > 16 \), the base case holds.

Step 2: Inductive Hypothesis

Assume that the statement \( n! > 2^n \) holds for some integer \( k \geq 4 \). This means \( k! > 2^k \).

Step 3: Inductive Step

We need to show that \( (k+1)! > 2^{k+1} \). Starting with \( (k+1)! = (k+1) imes k! \), use the inductive hypothesis: \( (k+1) imes k! > (k+1) imes 2^k \). Since \( k+1 \geq 5 \) for \( k \geq 4 \), it follows \( (k+1) imes 2^k \geq 2 \times 2^k = 2^{k+1} \). Therefore, \( (k+1)! > 2^{k+1} \).

Step 4: Conclusion: Inductive Proof Completion

By the Principle of Mathematical Induction, the inequality \( n! > 2^n \) holds for all integers \( n \geq 4 \). The base case is verified and each inductive step proves the statement for \( n+1 \) given it holds for \( n \).

Key Concepts

Factorial

The factorial of a number is an important concept in mathematics, particularly in permutations, combinations, and other areas relating to growth or multiplicity. Factorials are denoted by an exclamation mark (!). For example, the factorial of a number \( n \), written as \( n! \), is the product of all positive integers less than or equal to \( n \).

Here's how you compute a factorial:

• \( 1! = 1 \)
• \( 2! = 2 \times 1 = 2 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Factorials grow very quickly as the number increases, which is why in our inequality \( n! > 2^n \), \( n! \) starts to exceed \( 2^n \) as \( n \) gets larger.

They are used for counting permutations, determining coefficients in equations, and various proofs in mathematical theories.

Exponentiation

Exponentiation is the process of raising a number, known as the base, to the power of an exponent. It is a key operation in mathematics representing repeated multiplication of the base. Exponents are written as superscripts. For example, \( 2^n \) means multiplying 2 by itself \( n \) times.

Here's a quick overview of basic exponential values:

• \( 2^1 = 2 \)
• \( 2^2 = 4 \)
• \( 2^3 = 8 \)
• \( 2^4 = 16 \)

Despite being a rapid form of growth, exponential functions like \( 2^n \) initially increase at a slower rate compared to factorial functions, which is why for large \( n \), factorials surpass exponentials in value.

Exponentiation is a fundamental operation in various fields including algebra, computer science, and natural sciences.

Inequality

An inequality compares two values, showing if one quantity is less than, greater than, or not equal to the other. In mathematics, inequalities allow us to express relationships between expressions that are not exactly equal but still have a meaningful comparison.

In our problem, the inequality \( n! > 2^n \) suggests that beyond a certain point, the growth rate of \( n! \) surpasses that of \( 2^n \). Understanding inequalities helps us make estimations and predictions about numerical relationships and solve equations that do not have simple equality.

Inequalities are expressed using symbols like:

• \( > \) means greater than
• \( < \) means less than
• \( \geq \) means greater than or equal to
• \( \leq \) means less than or equal to

Mastery of inequalities is crucial for solving algebraic equations, modeling real-world situations, and understanding complex systems in science and engineering.

Proof Technique

Mathematical induction is a fundamental proof technique used to prove statements about all natural numbers. It consists of two main parts: the base case and the inductive step.

Here's how it works:

• Base Case: Verify the statement holds for an initial value, usually a small number like \( n = 4 \).
• Inductive Step: Assume the statement is true for a number \( k \), then prove it for \( k+1 \). If both parts hold, then the statement is true for all numbers starting from the base case.

In our proof for \( n! > 2^n \), we first checked it for \( n=4 \), and then assumed it was true for \( k \) to prove for \( k+1 \).

Mathematical induction is not only a powerful tool but also elegant — allowing us to prove properties of sequences, series, and other mathematical constructs.