A radioactive nucleus undergoes a series of decays according to the sequence :
`A overset(beta)rarr A_(1) overset(alpha)rarr A_(2) overset(alpha) rarr A_(3)`
If the mass number and atomic number of `A_(3)` are 172 and 69 respectively, what rate the mass number and atomic number of A ?

Answer

To solve the problem, we need to analyze the decay sequence of the radioactive nucleus and determine the mass number (A) and atomic number (Z) of the initial nucleus (A).

### Step-by-Step Solution:

1. **Understanding the Decay Sequence**:
- The initial nucleus is denoted as A.
- It undergoes beta decay to form A1.
- A1 then undergoes alpha decay to form A2.
- A2 undergoes another alpha decay to form A3.

2. **Identifying the Changes in Atomic Number and Mass Number**:
- **Beta Decay**:
- In beta decay, the atomic number (Z) increases by 1, while the mass number (A) remains unchanged.
- Therefore, if A has atomic number Z, then A1 will have atomic number Z + 1 and mass number A.

- **Alpha Decay**:
- In alpha decay, the atomic number decreases by 2 and the mass number decreases by 4.
- Therefore, if A1 has atomic number Z + 1 and mass number A, then A2 will have atomic number (Z + 1 - 2) = Z - 1 and mass number (A - 4).
- Continuing this, A3 will have atomic number (Z - 1 - 2) = Z - 3 and mass number (A - 4 - 4) = A - 8.

3. **Using the Given Information for A3**:
- We know that for A3:
- Atomic number (Z3) = 69
- Mass number (A3) = 172

4. **Setting Up the Equations**:
- From the decay sequence, we have:
- Z3 = Z - 3
- A3 = A - 8
- Substituting the values for A3:
- 69 = Z - 3 → Z = 69 + 3 = 72
- 172 = A - 8 → A = 172 + 8 = 180

5. **Final Result**:
- The mass number (A) of the initial nucleus A is 180.
- The atomic number (Z) of the initial nucleus A is 72.

### Conclusion:
- The mass number of A is 180, and the atomic number of A is 72.