To find the ratio of the velocities of the electron in the first, second, and third orbits (V1:V2:V3) for any hydrogen-like system, we can follow these steps:
### Step-by-Step Solution:
1. **Understanding the Relationship**:
The velocity of an electron in a hydrogen-like atom is given by the formula:
\[
V_n \propto \frac{Z}{n}
\]
where \( V_n \) is the velocity in the nth orbit, \( Z \) is the atomic number, and \( n \) is the principal quantum number (the orbit number).
2. **Identifying the Atomic Number**:
For hydrogen-like systems, the atomic number \( Z \) is constant for a given atom. Therefore, we can focus on the relationship between the velocities and the orbit numbers.
3. **Setting Up the Ratios**:
Since \( V_n \propto \frac{Z}{n} \), we can express the velocities for the first, second, and third orbits as:
\[
V_1 \propto \frac{Z}{1}, \quad V_2 \propto \frac{Z}{2}, \quad V_3 \propto \frac{Z}{3}
\]
4. **Forming the Velocity Ratio**:
The ratio of the velocities \( V_1 : V_2 : V_3 \) can be expressed as:
\[
V_1 : V_2 : V_3 = \frac{Z}{1} : \frac{Z}{2} : \frac{Z}{3}
\]
Since \( Z \) is common in all terms, it cancels out:
\[
V_1 : V_2 : V_3 = 1 : \frac{1}{2} : \frac{1}{3}
\]
5. **Finding a Common Denominator**:
To simplify this ratio, we can convert it to a common denominator:
\[
1 : \frac{1}{2} : \frac{1}{3} = 6 : 3 : 2
\]
6. **Final Ratio**:
Thus, the final ratio of velocities of the electron in the first, second, and third orbits is:
\[
V_1 : V_2 : V_3 = 6 : 3 : 2
\]
### Conclusion:
The ratio of the velocities of the electron in the first, second, and third orbits of a hydrogen-like system is \( 6 : 3 : 2 \).
