Proving the Inequality Using Mathematical Induction

Mathematical induction is a powerful proof technique used to demonstrate the validity of statements for all natural numbers. In this article, we will use mathematical induction to prove an inequality involving factorials and powers of 2. Specifically, we will prove that for any natural number ( n ), the following inequality holds:

For ( n 4 ), we have: 4! 24 16 2^4

We will proceed with a step-by-step proof, breaking it down into a clear and detailed process.

Step 1: Basis Step

First, we need to establish the base case, which is the initial value of ( n ). In this problem, the base case is ( n 4 ).

Base Case:

~For ( n 4 ), the left-hand side (LHS) is given by:

~LHS 4! 4 × 3 × 2 × 1 24

~The right-hand side (RHS) is given by:

~RHS 2^4 16

~Clearly, 24 is greater than 16, so the inequality holds for the base case.

Step 2: Inductive Step

Next, we need to prove that if the inequality holds for some ( n k ), it also holds for ( n k 1 ). This is the inductive step.

Assume that for ( n k ), the inequality holds, i.e.,

~

~We need to prove that

~(k 1)! ≥ 2^{k 1}

Inductive Hypothesis:

~Using the inductive hypothesis, we know:

~(k 1)! (k 1) × k!

~Since ( k! ≥ 2^k ) by the inductive hypothesis:

~({k 1} × k!) ≥ (k 1) × 2^k

~This simplifies to:

~{k 1} × 2^k ≥ 2^{k 1}

~To prove this, we need to show:

~{k 1} × 2^k ≥ 2 × 2^k

~Simplifying, we get:

~{k 1} ≥ 2

~Since ( k ) is a natural number greater than or equal to 4, ( k 1 ) is certainly greater than or equal to 2. Thus, the inequality holds.

Conclusion:

By the principle of mathematical induction, the inequality n! ≥ 2^n holds for all natural numbers ( n ).

Additional Considerations

If you are writing this as an exam question, ensure you include all the steps clearly and logically. Here’s a structured way to present it:

Basis Step: Verify the inequality for ( n 4 ). Inductive Step: Assume it holds for ( n k ), and prove it for ( n k 1 ). Conclusion: State the final result and indicate that the proof by induction is complete.

This method not only demonstrates the correctness of the inequality but also provides a clear and thorough understanding of the process.

Further, understanding the mechanism of induction can be applied to a variety of problems in mathematics and computer science, ultimately enhancing problem-solving skills.