In a radioactive sample, the fraction of initial number of redioactive nuclie, which remains undecayed after n mean lives is

Options

A. `(1)/(e^(n))`
B. `e^(n)`
C. `1-(1)/(e^(n))`
D. `((1)/(e-1))^(n)`

Explanation

To solve the problem of finding the fraction of the initial number of radioactive nuclei that remains undecayed after \( n \) mean lives, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Mean Life**:
The mean life (\( \tau \)) of a radioactive substance is the average time that a nucleus exists before it decays. It is related to the decay constant (\( \lambda \)) by the formula:
\[ \tau = \frac{1}{\lambda} \]

2. **Determine the Time After \( n \) Mean Lives**:
If we consider \( n \) mean lives, the total time \( t \) can be expressed as:
\[ t = n \cdot \tau = n \cdot \frac{1}{\lambda} = \frac{n}{\lambda} \]

3. **Use the Decay Formula**:
The number of undecayed nuclei \( N(t) \) after time \( t \) is given by the exponential decay formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of nuclei.

4. **Substitute for Time \( t \)**:
Substitute \( t = \frac{n}{\lambda} \) into the decay formula:
\[ N(t) = N_0 e^{-\lambda \left(\frac{n}{\lambda}\right)} = N_0 e^{-n} \]

5. **Find the Fraction of Undecayed Nuclei**:
The fraction of undecayed nuclei after \( n \) mean lives is given by:
\[ \text{Fraction} = \frac{N(t)}{N_0} = \frac{N_0 e^{-n}}{N_0} = e^{-n} \]

6. **Final Result**:
Therefore, the fraction of the initial number of radioactive nuclei that remains undecayed after \( n \) mean lives is:
\[ \text{Fraction} = \frac{1}{e^n} \]

### Conclusion:
The answer to the question is:
\[ \frac{1}{e^n} \]