To prove $2^n \geq n^2$ for all $n \geq 5$ using mathematical induction, we need to follow three steps: the base case, the induction hypothesis, and the induction step.

Step 1: Base Case ($n = 5$)

For $n = 5$: $2^5 = 32 \quad \text{and} \quad 5^2 = 25.$ Clearly, $32 \geq 25$, so the base case holds.

Step 2: Induction Hypothesis

Assume that $2^k \geq k^2$ is true for some integer $k \geq 5$. That is, we assume: $2^k \geq k^2.$

Step 3: Induction Step

We need to show that $2^{k+1} \geq (k+1)^2$ given that $2^k \geq k^2$.

Start with the left-hand side of the inequality: $2^{k+1} = 2 \cdot 2^k.$

By the induction hypothesis, we know $2^k \geq k^2$, so we substitute: $2^{k+1} \geq 2 \cdot k^2.$

Now, we need to show that: $2 \cdot k^2 \geq (k+1)^2.$

Expanding $(k+1)^2$: $(k+1)^2 = k^2 + 2k + 1.$

So, it is enough to show: $2k^2 \geq k^2 + 2k + 1.$

Simplify this inequality: $k^2 \geq 2k + 1.$

This can be rewritten as: $k^2 - 2k - 1 \geq 0.$

Step 4: Verify for $k \geq 5$

Now, we need to check if the quadratic $k^2 - 2k - 1$ is non-negative for $k \geq 5$.

The quadratic $k^2 - 2k - 1$ can be solved using the quadratic formula: $k = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}.$

The roots are approximately $1 - \sqrt{2} \approx -0.41$ and $1 + \sqrt{2} \approx 2.41$. Since $k \geq 5$, this inequality holds true for all $k \geq 5$.

Thus, $k^2 - 2k - 1 \geq 0$ for $k \geq 5$, which means: $2k^2 \geq (k+1)^2 \quad \text{for all } k \geq 5.$

Conclusion

By induction, we have shown that $2^n \geq n^2$ for all $n \geq 5$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the quadratic formula help solve inequalities in induction proofs?
2. What are some other examples where induction is useful in proving inequalities?
3. Can induction be used to prove geometric series inequalities?
4. How can we verify induction results using graphing techniques?
5. Why is the base case important in an induction proof?

Tip: When using induction, always ensure that the induction hypothesis is correctly assumed before moving to the next step.