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

\[
2^n \geq n^2 \quad \text{for every real number} \, n \geq 9
\]

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 \( n = 9 \).

\[
2^9 = 512, \quad 9^2 = 81
\]
Clearly, \( 512 \geq 81 \), so the base case \( n = 9 \) holds.

### Step 2: Inductive Hypothesis
Assume the statement is true for some \( n = k \geq 9 \), that is, assume:

\[
2^k \geq k^2
\]

### Step 3: Inductive Step
We now need to prove that the inequality holds for \( n = k + 1 \), i.e., prove that:

\[
2^{k+1} \geq (k+1)^2
\]

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

\[
2^{k+1} = 2 \cdot 2^k \geq 2 \cdot k^2
\]

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

\[
2k^2 \geq (k+1)^2 = k^2 + 2k + 1
\]

Thus, we need to prove:

\[
2k^2 \geq k^2 + 2k + 1
\]

This simplifies to:

\[
k^2 \geq 2k + 1
\]

This inequality holds for \( k \geq 9 \), as we can verify by plugging in values of \( k \).

### Conclusion:
Since the base case holds and the inductive step is proven, by mathematical induction, the inequality \( 2^n \geq n^2 \) holds for all \( n \geq 9 \).

---

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 \( n \)?
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.

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

Learn how to prove the inequality 2^n ≥ n^2 for every real number n ≥ 9 using mathematical induction. This step-by-step solution covers the base case, inductive hypothesis, and inductive step. Suitable for advanced high school students or undergraduate-level mathematics.

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