For each `n in N,` the correct statement is

Options

A. `2 ^(n)lt n`
B. `n ^(2) gt 2n`
C. `n ^(4) lt 10 ^(n)`
D. `2 ^(3n) gt 7n +1 `

Answer

To solve the problem, we need to analyze each of the given statements for \( n \in \mathbb{N} \) (natural numbers). Let's evaluate each statement step by step.

### Step-by-Step Solution

1. **Statement 1: \( 1^n < n \)**
- For \( n = 1 \):
\[ 1^1 < 1 \implies 1 < 1 \quad \text{(False)} \]
- For \( n = 2 \):
\[ 1^2 < 2 \implies 1 < 2 \quad \text{(True)} \]
- For \( n = 3 \):
\[ 1^3 < 3 \implies 1 < 3 \quad \text{(True)} \]
- However, since the statement must hold for all \( n \in \mathbb{N} \), and it is false for \( n = 1 \), this statement is **False**.

2. **Statement 2: \( n^2 > 2n \)**
- For \( n = 1 \):
\[ 1^2 > 2 \cdot 1 \implies 1 > 2 \quad \text{(False)} \]
- For \( n = 2 \):
\[ 2^2 > 2 \cdot 2 \implies 4 > 4 \quad \text{(False)} \]
- For \( n = 3 \):
\[ 3^2 > 2 \cdot 3 \implies 9 > 6 \quad \text{(True)} \]
- Since it is false for \( n = 1 \) and \( n = 2 \), this statement is also **False**.

3. **Statement 3: \( n^4 < 10^n \)**
- For \( n = 1 \):
\[ 1^4 < 10^1 \implies 1 < 10 \quad \text{(True)} \]
- For \( n = 2 \):
\[ 2^4 < 10^2 \implies 16 < 100 \quad \text{(True)} \]
- For \( n = 3 \):
\[ 3^4 < 10^3 \implies 81 < 1000 \quad \text{(True)} \]
- This statement holds true for \( n = 1, 2, 3 \) and will continue to hold for larger \( n \) since \( 10^n \) grows much faster than \( n^4 \). Thus, this statement is **True**.

4. **Statement 4: \( 2^{3n} > 7n + 1 \)**
- For \( n = 1 \):
\[ 2^{3 \cdot 1} > 7 \cdot 1 + 1 \implies 8 > 8 \quad \text{(False)} \]
- For \( n = 2 \):
\[ 2^{3 \cdot 2} > 7 \cdot 2 + 1 \implies 64 > 15 \quad \text{(True)} \]
- For \( n = 3 \):
\[ 2^{3 \cdot 3} > 7 \cdot 3 + 1 \implies 512 > 22 \quad \text{(True)} \]
- However, since it is false for \( n = 1 \), this statement is also **False**.

### Conclusion
The only true statement among the four is:
\[ n^4 < 10^n \quad \text{for all } n \in \mathbb{N} \]