Problem 27 - College Algebra with Modeling and Visualization

Question

Prove the statement by mathematical induction. If \(n \geq 4,\) then \(n !>2^{n},\) where \(n !=n(n-1)(n-2) \cdots(3)(2)(1)\)

Step-by-Step Solution

Answer

By mathematical induction, we proved that when \(n \geq 4, n! > 2^n.\)

Step 1: Base Case Verification

Start by checking the smallest value of \(n\), which is 4.Calculate \(n!\) for \(n=4\): \[4! = 4 \times 3 \times 2 \times 1 = 24\]Calculate \(2^n\) for \(n=4\): \[2^{4} = 16\]Since \(24 > 16\), the base case holds true.

Step 2: Inductive Hypothesis

Assume that the statement is true for some arbitrary positive integer \(k\geq4\). That is, assume \[ k! > 2^k \]This is our inductive hypothesis and we will use it to prove the next case.

Step 3: Inductive Step

We need to prove that the statement holds for \(k + 1\), assuming it is true for \(k\). Start with the expression for \((k+1)!\): \[(k+1)! = (k+1) \times k!\]Using the inductive hypothesis: \[ (k+1) \times k! > (k+1) \times 2^k \]Also, note that since \(k\geq4\), we have \(k+1 \geq 5\), and thus:\[ (k+1) \times 2^k > 2 \times 2^k = 2^{k+1} \]Hence,\[(k+1)! > 2^{k+1}\]This completes the inductive step.

Key Concepts

factorials

Factorials are a fundamental concept in mathematics, often denoted by the symbol '!'. If you've ever wondered how quickly numbers can grow, factorials are a great demonstration! The factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\). In mathematical terms, it is given by:

• \(n! = n \times (n-1) \times (n-2) \cdots \times 3 \times 2 \times 1\)

Factorials are used in a wide range of mathematical disciplines, including combinatorics, algebra, and calculus. One key feature is their rapid growth rate compared to other functions like exponential functions. For example, as shown in the original exercise for \(n = 4\), \(4! = 24\). This already outpaces \(2^4 = 16\), which illustrates how quickly factorials rise.

inequalities

Inequalities are expressions that show the relationship where one quantity is larger or smaller than another. They are a core concept in mathematics, often involving symbols such as:

• '>' for greater than
• '<' for less than
• '≥' for greater than or equal to
• '≤' for less than or equal to

In the exercise, the goal is to demonstrate that the factorial \(n!\) is greater than the exponential function \(2^n\) for \(n \geq 4\). Recognizing and working with inequalities can help solve complex problems by assessing bounds and limits. You often need to transform these expressions to develop proofs or solutions. This transformation was significant in the inductive step where we showed \((k+1)! > 2^{k+1}\), ensuring that the factorial increases faster than the exponential function.

inductive hypothesis

The inductive hypothesis is a crucial step in the process of mathematical induction, which mathematicians use to prove that a statement is true for all natural numbers. Think of it like climbing a ladder: if you're at a rung \(k\), and you can prove that you can step up to rung \(k+1\), then you can climb infinitely. Here's how it works:First, confirm the base case to ensure that the statement is true for the initial rung \(n\). For the given exercise, that was \(n = 4\).Once you verify this, the inductive hypothesis assumes that the statement \(n! > 2^n\) is true for some integer \(k\).Next, you use this assumption to prove that the statement must also be true for \(k+1\). This involves showing that \((k+1)! > 2^{k+1}\), completing the inductive step.The inductive hypothesis is like the backbone of mathematical induction, allowing us to extend a verified truth across an infinite set of numbers. By mastering this approach, you unlock a powerful technique for establishing broader truths from smaller ones.