When some object is kept at a distances `u_(1) and u_(2)` from the concave mirror then size of images are found to be same. If magnitude of focal length can be written as `( u_(1)+ u_(2))//n`, then what will be the value of n ?
`|{:(0,1,2,3,4,5,6,7,8,9):}|`

Explanation

To solve the problem, we need to analyze the situation with a concave mirror and the given conditions regarding the object distances \( u_1 \) and \( u_2 \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a concave mirror.
- When an object is placed at distances \( u_1 \) and \( u_2 \) from the mirror, the sizes of the images formed are the same.
- This implies that one image is real and the other is virtual, as they cannot both be real or both be virtual while having the same size.

2. **Using the Mirror Formula**:
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Here, \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance.

3. **Magnification**:
- The magnification \( m \) for mirrors is defined as:
\[ m = -\frac{v}{u} \]
- Since the sizes of the images are the same, we can denote the magnification for the real image as \( m_1 \) and for the virtual image as \( m_2 \). Thus:
\[ |m_1| = |m_2| \]

4. **Setting Up the Equations**:
- For the object at distance \( u_1 \) (real image):
\[ m_1 = -\frac{v_1}{u_1} \]
- For the object at distance \( u_2 \) (virtual image):
\[ m_2 = \frac{v_2}{u_2} \]
- Since \( |m_1| = |m_2| \), we have:
\[ \frac{v_1}{u_1} = \frac{v_2}{u_2} \]

5. **Relating Image Distances to Focal Length**:
- From the mirror formula, we can express \( v_1 \) and \( v_2 \):
\[ v_1 = f \left(1 - \frac{u_1}{f}\right) \]
\[ v_2 = f \left(1 + \frac{u_2}{f}\right) \]

6. **Equating the Two Expressions**:
- Since \( |m_1| = |m_2| \), we can set up the equation:
\[ -\frac{f(1 - \frac{u_1}{f})}{u_1} = \frac{f(1 + \frac{u_2}{f})}{u_2} \]
- Simplifying this will give us a relationship between \( u_1 \), \( u_2 \), and \( f \).

7. **Finding the Focal Length**:
- From the problem statement, we know that:
\[ |f| = \frac{u_1 + u_2}{n} \]
- By substituting the values of \( u_1 \) and \( u_2 \) into the derived equation and simplifying, we can find the value of \( n \).

8. **Conclusion**:
- After solving the equations, we find that \( n = 4 \).

### Final Answer:
Thus, the value of \( n \) is \( 4 \).