Choose the proposition that is not true for `n gt 1 (n in N)`.

Options

A. `1+(1)/(4)+(1)/(9)+...+(1)/(n^(2))lt 2 -(1)/(n)`
B. `1+(1)/(sqrt2)+(1)/(sqrt3)+...+(1)/(sqrtn)gt sqrtn`
C. `(1)/(n+1)+(1)/(n+2)+....+(1)/(2n)gt (13)/(24)`
D. `(1)/(2).(3)/(4).(5)/(6)...+(2n-1)/(2n)gt(1)/(sqrt(3n+1))`

Answer

To solve the problem, we need to evaluate each proposition for \( n > 1 \) where \( n \) is a natural number. Let's check each option step by step.

### Step 1: Evaluate Option 1
**Proposition:** \( 1 + \frac{1}{4} n^2 < n - \frac{1}{2} \)

1. Substitute \( n = 2 \):
\[ 1 + \frac{1}{4} \cdot 2^2 < 2 - \frac{1}{2} \]
This simplifies to:
\[ 1 + \frac{1}{4} \cdot 4 < 2 - 0.5 \]
\[ 1 + 1 < 1.5 \]
\[ 2 < 1.5 \quad \text{(False)} \]

**Conclusion:** This proposition is not true for \( n = 2 \).

### Step 2: Evaluate Option 2
**Proposition:** \( 1 + \frac{1}{\sqrt{n}} > \sqrt{n} \)

1. Substitute \( n = 2 \):
\[ 1 + \frac{1}{\sqrt{2}} > \sqrt{2} \]
This simplifies to:
\[ 1 + \frac{1}{1.414} > 1.414 \]
\[ 1 + 0.707 > 1.414 \]
\[ 1.707 > 1.414 \quad \text{(True)} \]

### Step 3: Evaluate Option 3
**Proposition:** \( \frac{1}{2} + \frac{1}{n} > \frac{13}{24} \)

1. Substitute \( n = 2 \):
\[ \frac{1}{2} + \frac{1}{2} > \frac{13}{24} \]
This simplifies to:
\[ 1 > \frac{13}{24} \]
\[ 1 = \frac{24}{24} > \frac{13}{24} \quad \text{(True)} \]

### Step 4: Evaluate Option 4
**Proposition:** \( \frac{1}{2} \cdot \frac{3}{4} > \frac{1}{6} + \sqrt{n} \)

1. Substitute \( n = 2 \):
\[ \frac{3}{8} > \frac{1}{6} + \sqrt{2} \]
This simplifies to:
\[ 0.375 > 0.1667 + 1.414 \]
\[ 0.375 > 1.5807 \quad \text{(False)} \]

### Final Conclusion
The proposition that is not true for \( n > 1 \) is **Option 4**.

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