If `N^(0)` is the initial number of nuclei, number of nuclei remaining undecayed at the end of `n^(th)` half-life is

Options

A. `2^(-n) N^(0)`
B. `2^(n) N^(0)`
C. `n^(-2) N^(0)`
D. `n^(2) N^(0)`

Answer

To solve the problem of determining the number of undecayed nuclei remaining after the end of the Nth half-life, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
- The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. After one half-life, half of the original nuclei remain.

2. **Initial Number of Nuclei**:
- Let \( N_0 \) be the initial number of nuclei.

3. **Determine the Number of Nuclei After Each Half-Life**:
- After the first half-life, the number of undecayed nuclei will be:
\[ N_1 = \frac{N_0}{2} \]
- After the second half-life, the number of undecayed nuclei will be:
\[ N_2 = \frac{N_1}{2} = \frac{N_0}{2^2} \]
- After the third half-life, the number of undecayed nuclei will be:
\[ N_3 = \frac{N_2}{2} = \frac{N_0}{2^3} \]
- Continuing this pattern, after \( n \) half-lives, the number of undecayed nuclei will be:
\[ N_n = \frac{N_0}{2^n} \]

4. **Final Expression**:
- Therefore, the number of nuclei remaining undecayed at the end of the Nth half-life is given by:
\[ N_n = N_0 \cdot 2^{-n} \]

### Conclusion:
The number of undecayed nuclei remaining after \( n \) half-lives is \( N_0 \cdot 2^{-n} \).