Understanding n! ≥ 2^(n-1) with Induction

The inequality n! ≥ 2^n-1 is an interesting mathematical result that shows how rapidly the factorial function grows compared to an exponential function. In this blog, we will prove this inequality using the method of mathematical induction. This is a great way to understand the behavior of factorials and exponentials for positive integers.

Understanding the Components

Before diving into the proof, let’s briefly review the two functions involved:

• Factorial (n!): The factorial of n is the product of all positive integers from 1 to n. For example, 4! = 4 × 3 × 2 × 1 = 24.
• Exponential (2^n-1): The exponential function with base 2 grows by doubling with each increase in n. For example, if n = 4, then 2^n-1 = 2³ = 8.

We aim to prove that n! ≥ 2^n-1 for all integers n ≥ 1.

The Proof Using Mathematical Induction

Mathematical induction is a method of proving statements that hold true for all positive integers. It involves two key steps:

1. Base Case: Verify the inequality for the smallest value of n.
2. Inductive Step: Assume the statement holds for some n = k, and prove it holds for n = k+1.

Base Case (n = 1)

For n = 1:

1! = 1 and 2^1-1 = 2⁰ = 1.

Clearly, 1! ≥ 2^1-1. Thus, the base case holds true.

Inductive Hypothesis

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

k! ≥ 2^k-1.

We need to prove that the inequality also holds for n = k+1, i.e.,

(k+1)! ≥ 2^k.

Inductive Step

Start with the definition of factorial:

(k+1)! = (k+1) × k!.

Using the inductive hypothesis k! ≥ 2^k-1, substitute k!:

(k+1)! ≥ (k+1) × 2^k-1.

We need to show that:

(k+1) × 2^k-1 ≥ 2^k.

Simplify the right-hand side by writing 2^k as 2 × 2^k-1:

(k+1) × 2^k-1 ≥ 2 × 2^k-1.

Divide both sides by 2^k-1 (which is positive):

k+1 ≥ 2.

This inequality is true for all k ≥ 1.

Thus, the inequality holds for n = k+1.

Conclusion

By the principle of mathematical induction, we have proven that:

n! ≥ 2^n-1 for all integers n ≥ 1.

Why This Inequality Matters

This inequality is an excellent demonstration of how factorials grow much faster than exponential functions. It has practical applications in combinatorics, algorithm analysis, and mathematical modeling, where understanding growth rates is essential.

If you have any thoughts or questions about this proof, feel free to share them in the comments below!