The statement `2^(n) ge n^(2)` (where `n in N)` is true for

Options

A. All n
B. `n ge 3`
C. `n lt 4`
D. `n gt 4`

Answer

To determine for which values of \( n \in \mathbb{N} \) the statement \( 2^n \geq n^2 \) holds true, we can evaluate the expression for various natural numbers.

**Step 1: Check for \( n = 1 \)**
- Calculate \( 2^1 \) and \( 1^2 \):
\[ 2^1 = 2 \quad \text{and} \quad 1^2 = 1 \]
- Compare:
\[ 2 \geq 1 \quad \text{(True)} \]

**Step 2: Check for \( n = 2 \)**
- Calculate \( 2^2 \) and \( 2^2 \):
\[ 2^2 = 4 \quad \text{and} \quad 2^2 = 4 \]
- Compare:
\[ 4 \geq 4 \quad \text{(True)} \]

**Step 3: Check for \( n = 3 \)**
- Calculate \( 2^3 \) and \( 3^2 \):
\[ 2^3 = 8 \quad \text{and} \quad 3^2 = 9 \]
- Compare:
\[ 8 \geq 9 \quad \text{(False)} \]

**Step 4: Check for \( n = 4 \)**
- Calculate \( 2^4 \) and \( 4^2 \):
\[ 2^4 = 16 \quad \text{and} \quad 4^2 = 16 \]
- Compare:
\[ 16 \geq 16 \quad \text{(True)} \]

**Step 5: Check for \( n = 5 \)**
- Calculate \( 2^5 \) and \( 5^2 \):
\[ 2^5 = 32 \quad \text{and} \quad 5^2 = 25 \]
- Compare:
\[ 32 \geq 25 \quad \text{(True)} \]

**Step 6: Check for \( n = 6 \)**
- Calculate \( 2^6 \) and \( 6^2 \):
\[ 2^6 = 64 \quad \text{and} \quad 6^2 = 36 \]
- Compare:
\[ 64 \geq 36 \quad \text{(True)} \]

**Step 7: Check for \( n = 7 \)**
- Calculate \( 2^7 \) and \( 7^2 \):
\[ 2^7 = 128 \quad \text{and} \quad 7^2 = 49 \]
- Compare:
\[ 128 \geq 49 \quad \text{(True)} \]

**Conclusion:**
From the evaluations, we see that the statement \( 2^n \geq n^2 \) is true for \( n = 1, 2, 4, 5, 6, 7 \), and it fails for \( n = 3 \).

To determine the range of \( n \) for which the statement is true, we observe that:
- The statement is true for \( n = 1, 2 \).
- The statement becomes false at \( n = 3 \).
- The statement is true for \( n \geq 4 \).

Thus, the correct answer is that the statement \( 2^n \geq n^2 \) is true for \( n \geq 4 \).

### Final Answer:
The statement \( 2^n \geq n^2 \) is true for \( n \geq 4 \).

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