Let `P(n): (2n+1) lt 2^(n)`, then the smallest positive integer for which P(n) is true?

Options

A. (a)2
B. (b)3
C. (c)4
D. (d)5

Answer

To solve the problem, we need to find the smallest positive integer \( n \) such that the inequality \( P(n): 2n + 1 < 2^n \) holds true. We will use a trial-and-error method to find the smallest integer that satisfies this inequality.

### Step-by-Step Solution:

1. **Start with \( n = 1 \)**:
- Calculate the left-hand side (LHS):
\[ LHS = 2(1) + 1 = 2 + 1 = 3 \]
- Calculate the right-hand side (RHS):
\[ RHS = 2^1 = 2 \]
- Check the inequality:
\[ 3 < 2 \quad \text{(False)} \]
- Conclusion: \( n = 1 \) does not satisfy \( P(n) \).

2. **Next, try \( n = 2 \)**:
- Calculate the LHS:
\[ LHS = 2(2) + 1 = 4 + 1 = 5 \]
- Calculate the RHS:
\[ RHS = 2^2 = 4 \]
- Check the inequality:
\[ 5 < 4 \quad \text{(False)} \]
- Conclusion: \( n = 2 \) does not satisfy \( P(n) \).

3. **Now, try \( n = 3 \)**:
- Calculate the LHS:
\[ LHS = 2(3) + 1 = 6 + 1 = 7 \]
- Calculate the RHS:
\[ RHS = 2^3 = 8 \]
- Check the inequality:
\[ 7 < 8 \quad \text{(True)} \]
- Conclusion: \( n = 3 \) satisfies \( P(n) \).

4. **Verify if there is a smaller integer**:
- Since we have already checked \( n = 1 \) and \( n = 2 \) and found that they do not satisfy the inequality, we confirm that \( n = 3 \) is indeed the smallest positive integer for which \( P(n) \) is true.

### Final Answer:
The smallest positive integer \( n \) for which \( P(n) \) is true is \( n = 3 \).