To solve the inequality \( n! \geq 2^n - 1 \), let's analyze it step by step for different values of \( n \), starting from small numbers, and then find a general conclusion.

### Step-by-step approach:

1. **For \( n = 1 \):**
   \[
   1! = 1 \quad \text{and} \quad 2^1 - 1 = 1
   \]
   So, \( 1! \geq 2^1 - 1 \), which is true.

2. **For \( n = 2 \):**
   \[
   2! = 2 \quad \text{and} \quad 2^2 - 1 = 3
   \]
  

This detailed solution analyzes the inequality n! >= 2^n - 1 through a step-by-step approach, exploring factorial and exponential functions. We test small values of n and conclude a general solution. Ideal for students in Grades 10-12 looking to deepen their understanding of inequalities and factorials.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation