Three concentric spherical metallic shells A,B and C of radii a,b and c `( a lt b lt c)` have surface charge densities ` sigma, - sigma and sigma` respectively.
If the potenital of sheel B is `V_(B) = (sigma)/(in_(0)) ((a^(n))/(b) - b+c)` and the potential of shell C is `V_(C) = (sigma)/(in_(0)) ((a^(n))/(c) - (b^(n))/(c)+c)` then n is .

Explanation

To solve the problem, we need to find the value of \( n \) given the potentials \( V_B \) and \( V_C \) of the concentric spherical metallic shells A, B, and C.

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have three concentric spherical shells A, B, and C with radii \( a \), \( b \), and \( c \) respectively, where \( a < b < c \). The surface charge densities are \( \sigma \) for shell A, \( -\sigma \) for shell B, and \( \sigma \) for shell C.

2. **Potential of Shell B**:
The potential \( V_B \) is given by:
\[ V_B = \frac{\sigma}{\epsilon_0} \left( \frac{a^n}{b} - b + c \right) \]

3. **Potential of Shell C**:
The potential \( V_C \) is given by:
\[ V_C = \frac{\sigma}{\epsilon_0} \left( \frac{a^n}{c} - \frac{b^n}{c} + c \right) \]

4. **Finding the Charge Contributions**:
- The potential \( V_B \) is influenced by the charge on shell A and shell C (since shell B has a negative charge, it contributes negatively).
- The potential \( V_C \) is influenced by the charges on shells A and B.

5. **Expressing Potentials**:
We can express both potentials in terms of the charge densities and the radii:
- For shell B:
\[ V_B = \frac{1}{4\pi\epsilon_0} \left( \sigma \cdot 4\pi a^2 \cdot \frac{1}{b} - (-\sigma) \cdot 4\pi b^2 \cdot \frac{1}{b} + \sigma \cdot 4\pi c^2 \cdot \frac{1}{b} \right) \]
- For shell C:
\[ V_C = \frac{1}{4\pi\epsilon_0} \left( \sigma \cdot 4\pi a^2 \cdot \frac{1}{c} - (-\sigma) \cdot 4\pi b^2 \cdot \frac{1}{c} + \sigma \cdot 4\pi c^2 \cdot \frac{1}{c} \right) \]

6. **Comparing Potentials**:
To find \( n \), we need to compare the expressions for \( V_B \) and \( V_C \). We can equate the coefficients of similar terms in both equations.

7. **Finding n**:
By comparing the terms in \( V_B \) and \( V_C \):
- The term \( \frac{a^n}{b} \) in \( V_B \) corresponds to \( \frac{a^n}{c} \) in \( V_C \).
- The term \( -b + c \) in \( V_B \) corresponds to \( -\frac{b^n}{c} + c \) in \( V_C \).

From the comparison, we can derive that \( n = 2 \).

### Conclusion:
Thus, the value of \( n \) is:
\[ \boxed{2} \]