For every positive integral value of `n, 3 ^(n) gt n^(3)` when

Options

A. `n gt 2`
B. `n ge 2`
C. `n ge4`
D. `n lt 4`

Answer

To solve the inequality \(3^n > n^3\) for every positive integral value of \(n\), we can analyze the inequality step by step.

### Step 1: Understand the inequality
We need to determine for which values of \(n\) the expression \(3^n\) is greater than \(n^3\).

### Step 2: Test small values of \(n\)
Let's evaluate the inequality for small positive integers:

- **For \(n = 1\)**:
\[ 3^1 = 3 \quad \text{and} \quad 1^3 = 1 \quad \Rightarrow \quad 3 > 1 \quad \text{(True)} \]

- **For \(n = 2\)**:
\[ 3^2 = 9 \quad \text{and} \quad 2^3 = 8 \quad \Rightarrow \quad 9 > 8 \quad \text{(True)} \]

- **For \(n = 3\)**:
\[ 3^3 = 27 \quad \text{and} \quad 3^3 = 27 \quad \Rightarrow \quad 27 > 27 \quad \text{(False)} \]

- **For \(n = 4\)**:
\[ 3^4 = 81 \quad \text{and} \quad 4^3 = 64 \quad \Rightarrow \quad 81 > 64 \quad \text{(True)} \]

- **For \(n = 5\)**:
\[ 3^5 = 243 \quad \text{and} \quad 5^3 = 125 \quad \Rightarrow \quad 243 > 125 \quad \text{(True)} \]

### Step 3: Analyze the pattern
From our calculations:
- The inequality holds for \(n = 1\), \(n = 2\), and \(n = 4\) onwards.
- It fails for \(n = 3\).

### Step 4: General observation
As \(n\) increases beyond 4, \(3^n\) grows exponentially while \(n^3\) grows polynomially. Therefore, we can infer that for \(n \geq 4\), \(3^n\) will always be greater than \(n^3\).

### Conclusion
The inequality \(3^n > n^3\) holds true for every positive integral value of \(n\) when \(n \geq 4\).

### Final Answer
The answer is \(n \geq 4\).

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