Which of the following is true for `n in N`?

Options

A. `(1)/(2)+(1)/(4)+(1)/(8)....+(1)/(2^(n))lt 1`
B. `(1)/(n+1)+(1)/(n+2)+....+(1)/(3n+1)lt 1`
C. `n^(4) gt 10^(n)`
D. `.^(2n)C_(n)gt (4^(n))/(n+1)`

Answer

To solve the question "Which of the following is true for n in N?", we need to analyze each option provided in the question. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Understanding the Question**:
The question asks us to determine which of the given options is true for all natural numbers \( n \). Natural numbers are defined as \( N = \{1, 2, 3, \ldots\} \).

2. **Analyzing Option 1**:
The first option states a geometric progression (GP) with the first term \( a = \frac{1}{2} \) and common ratio \( r = \frac{1}{2} \). The sum of the first \( n \) terms of a GP is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Substituting the values:
\[ S_n = \frac{\frac{1}{2}(1 - (\frac{1}{2})^n)}{1 - \frac{1}{2}} = \frac{\frac{1}{2}(1 - \frac{1}{2^n})}{\frac{1}{2}} = 1 - \frac{1}{2^n} \]
Since \( n \) is a natural number, \( \frac{1}{2^n} \) is always positive, and thus \( S_n < 1 \) for all \( n \in N \). Therefore, this option is true.

3. **Analyzing Option 2**:
The second option states \( 1 + \frac{1}{1 + 2} < 1 \). Simplifying:
\[ 1 + \frac{1}{3} = 1.3333 \ldots \]
This is clearly not less than 1, so this option is false.

4. **Analyzing Option 3**:
The third option states \( 5^4 > 10^1 \). Calculating:
\[ 5^4 = 625 \quad \text{and} \quad 10^1 = 10 \]
This is true, but we need to check if it holds for all natural numbers. For \( n = 1 \), \( 5^1 = 5 \) which is not greater than \( 10^1 = 10 \). Hence, this option is false.

5. **Analyzing Option 4**:
The fourth option states \( 4 > 2 \) which is true, but we need to check if it holds for all natural numbers. For \( n = 1 \), \( 2 > 2 \) is false, so this option is also false.

### Conclusion:
Only option 1 is true for all natural numbers \( n \).

### Final Answer:
**Option 1 is the only true statement for \( n \in N \).**