The half line of a radioactive element is `2n` year. The fraction decayed in `n` year.

Options

A. `0.10`
B. `0.29`
C. `1.414`
D. `0.414`

Explanation

To solve the problem, we need to determine the fraction of a radioactive element that has decayed after a time period of `n` years, given that its half-life is `2n` years.

### Step-by-Step Solution:

1. **Understand the Half-Life Concept**:
The half-life of a radioactive element is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as `2n` years.

2. **Determine the Decay Formula**:
The amount of substance remaining after a certain time can be calculated using the formula:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
where:
- \( N \) = remaining quantity after time \( t \)
- \( N_0 \) = initial quantity
- \( T_{1/2} \) = half-life
- \( t \) = time elapsed

3. **Substituting Values**:
Here, \( T_{1/2} = 2n \) and \( t = n \). Substituting these values into the formula gives:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{n}{2n}} = N_0 \left( \frac{1}{2} \right)^{\frac{1}{2}} = N_0 \cdot \frac{1}{\sqrt{2}} \]

4. **Calculate the Remaining Quantity**:
From the above, we find that after `n` years, the remaining quantity \( N \) is:
\[ N = \frac{N_0}{\sqrt{2}} \]

5. **Calculate the Fraction Decayed**:
The fraction decayed can be calculated as:
\[ \text{Fraction Decayed} = \frac{N_0 - N}{N_0} = \frac{N_0 - \frac{N_0}{\sqrt{2}}}{N_0} \]
Simplifying this gives:
\[ \text{Fraction Decayed} = 1 - \frac{1}{\sqrt{2}} = 1 - \frac{\sqrt{2}}{2} \]

6. **Final Calculation**:
To get a numerical value, we can calculate:
\[ 1 - \frac{1}{\sqrt{2}} = 1 - 0.7071 \approx 0.2929 \]
Rounding this, we find that the fraction decayed is approximately:
\[ \text{Fraction Decayed} \approx 0.29 \]

### Final Answer:
The fraction decayed after `n` years is approximately **0.29**.