with induction, prove 3^(n) > n^4 for all integers n > 7

To prove that \(3^n > n^4\) for all integers \(n > 7\) using mathematical induction, we follow these steps:

### Step 1: Base Case
First, we verify the statement for \(n = 8\):

\[
3^8 = 6561
\]
\[
8^4 = 4096
\]

Clearly,
\[
6561 > 4096
\]

So, the base case holds true for \(n = 8\).

### Step 2: Inductive Hypothesis
Assume that for some integer \(k \geq 8\), the statement \(3^k > k^4\) holds true. That is,
\[
3^k > k^4
\]

### Step 3: Inductive Step
We need to show that \(3^{k+1} > (k+1)^4\).

Starting with the left-hand side:
\[
3^{k+1} = 3 \cdot 3^k
\]

Using the inductive hypothesis \(3^k > k^4\):
\[
3^{k+1} = 3 \cdot 3^k > 3 \cdot k^4
\]

Now, we need to show that \(3 \cdot k^4 \geq (k+1)^4\) for \(k \geq 8\):

\[
3 \cdot k^4 \geq (k+1)^4
\]

Expanding \((k+1)^4\):
\[
(k+1)^4 = k^4 + 4k^3 + 6k^2 + 4k + 1
\]

We need to show:
\[
3k^4 \geq k^4 + 4k^3 + 6k^2 + 4k + 1
\]

Subtract \(k^4\) from both sides:
\[
2k^4 \geq 4k^3 + 6k^2 + 4k + 1
\]

We need to verify if this inequality holds for \(k \geq 8\). For \(k = 8\):
\[
2 \cdot 8^4 = 2 \cdot 4096 = 8192
\]
\[
4 \cdot 8^3 = 4 \cdot 512 = 2048
\]
\[
6 \cdot 8^2 = 6 \cdot 64 = 384
\]
\[
4 \cdot 8 = 32
\]

Thus,
\[
4k^3 + 6k^2 + 4k + 1 \leq 2048 + 384 + 32 + 1 = 2465
\]

Clearly,
\[
8192 > 2465
\]

Therefore, \(2k^4 \geq 4k^3 + 6k^2 + 4k + 1\) holds for \(k = 8\). Since \(k^4\), \(4k^3\), \(6k^2\), \(4k\), and the constant 1 all grow polynomially and 2\(k^4\) grows faster than the polynomial sum as \(k\) increases, the inequality \(2k^4 \geq 4k^3 + 6k^2 + 4k + 1\) will hold for all \(k \geq 8\).

Thus, by induction, we have proven that \(3^n > n^4\) for all integers \(n > 7\).

Would you like more details or have any questions? Here are some more questions you might consider:

1. How do you prove the base case in an induction proof?
2. What are some common pitfalls in mathematical induction?
3. How does the inductive hypothesis work in a proof?
4. What is the importance of the inductive step?
5. Can induction be used to prove inequalities in general?

**Tip:** Always carefully check your base case and ensure your inductive step correctly applies the inductive hypothesis.

Learn how to prove the inequality 3^n > n^4 for all integers n > 7 using mathematical induction. This problem involves understanding mathematical induction, exponential growth, and inequalities. Suitable for advanced high school students.

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