Let `P(n) :n^(2) lt 2^(n), n gt 1`, then the smallest positive integer for which P(n) is true?
(i) 2
(ii) 3
(iii) 4
(iv) 5

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to determine the smallest positive integer \( n \) such that \( P(n) : n^2 < 2^n \) holds true for \( n > 1 \). We will check the values of \( n = 2, 3, 4, \) and \( 5 \).

### Step 1: Check \( n = 2 \)
- Calculate \( n^2 \):
\[ 2^2 = 4 \]
- Calculate \( 2^n \):
\[ 2^2 = 4 \]
- Compare:
\[ 4 < 4 \quad \text{(False)} \]

### Step 2: Check \( n = 3 \)
- Calculate \( n^2 \):
\[ 3^2 = 9 \]
- Calculate \( 2^n \):
\[ 2^3 = 8 \]
- Compare:
\[ 9 < 8 \quad \text{(False)} \]

### Step 3: Check \( n = 4 \)
- Calculate \( n^2 \):
\[ 4^2 = 16 \]
- Calculate \( 2^n \):
\[ 2^4 = 16 \]
- Compare:
\[ 16 < 16 \quad \text{(False)} \]

### Step 4: Check \( n = 5 \)
- Calculate \( n^2 \):
\[ 5^2 = 25 \]
- Calculate \( 2^n \):
\[ 2^5 = 32 \]
- Compare:
\[ 25 < 32 \quad \text{(True)} \]

### Conclusion
The smallest positive integer \( n \) for which \( P(n) \) is true is \( n = 5 \). Therefore, the answer is option (iv) 5.

---