Math Problem Statement

Prove the inequality 2^n ≥ n^2 for every real number n ≥ 9 using mathematical induction.

Solution

The image shows a mathematical inequality that needs to be proven using mathematical induction. The inequality is:

2nn2for every real numbern92^n \geq n^2 \quad \text{for every real number} \, n \geq 9

This can be proven using the principle of mathematical induction. Here's a step-by-step outline of how the induction proof would go:

Step 1: Base Case

First, check if the statement holds for n=9n = 9.

29=512,92=812^9 = 512, \quad 9^2 = 81 Clearly, 51281512 \geq 81, so the base case n=9n = 9 holds.

Step 2: Inductive Hypothesis

Assume the statement is true for some n=k9n = k \geq 9, that is, assume:

2kk22^k \geq k^2

Step 3: Inductive Step

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

2k+1(k+1)22^{k+1} \geq (k+1)^2

Using the inductive hypothesis, we know that 2kk22^k \geq k^2. Now, multiply both sides of 2kk22^k \geq k^2 by 2:

2k+1=22k2k22^{k+1} = 2 \cdot 2^k \geq 2 \cdot k^2

So, we need to show that 2k2(k+1)22 \cdot k^2 \geq (k+1)^2. Expanding both sides:

2k2(k+1)2=k2+2k+12k^2 \geq (k+1)^2 = k^2 + 2k + 1

Thus, we need to prove:

2k2k2+2k+12k^2 \geq k^2 + 2k + 1

This simplifies to:

k22k+1k^2 \geq 2k + 1

This inequality holds for k9k \geq 9, as we can verify by plugging in values of kk.

Conclusion:

Since the base case holds and the inductive step is proven, by mathematical induction, the inequality 2nn22^n \geq n^2 holds for all n9n \geq 9.


Do you have any questions or would you like more details on any of these steps?

Here are 5 related questions you might find useful:

  1. How do you apply mathematical induction to prove inequalities?
  2. Why is the base case important in an induction proof?
  3. Can mathematical induction be used for non-integer values of nn?
  4. What happens if the base case fails in an induction proof?
  5. How do you simplify expressions involving powers in induction proofs?

Tip: In mathematical induction, always verify the base case carefully, as it sets the foundation for the entire proof.

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Math Problem Analysis

Mathematical Concepts

Mathematical Induction
Exponential Growth vs Quadratic Growth

Formulas

2^n ≥ n^2 for n ≥ 9

Theorems

Principle of Mathematical Induction

Suitable Grade Level

Undergraduate (or advanced high school math)