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 potential of shell 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 \( n \) based on the given potentials of the spherical shells B and C. Let's break down the solution step by step.

### Step 1: Understand the Setup
We have three concentric spherical metallic 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.

### Step 2: Write the Expressions for Charge
The total charge on each shell can be expressed as:
- Charge on shell A, \( Q_A = \sigma \cdot 4\pi a^2 \)
- Charge on shell B, \( Q_B = -\sigma \cdot 4\pi b^2 \)
- Charge on shell C, \( Q_C = \sigma \cdot 4\pi c^2 \)

### Step 3: Write the Potential of Shell B
The potential \( V_B \) at shell B due to the charges inside it (from shells A and B) is given by:
\[ V_B = k \left( \frac{Q_A}{b} + \frac{Q_B}{b} \right) \]
Substituting the expressions for \( Q_A \) and \( Q_B \):
\[ V_B = k \left( \frac{\sigma \cdot 4\pi a^2}{b} - \frac{\sigma \cdot 4\pi b^2}{b} \right) \]
Simplifying this:
\[ V_B = k \left( \frac{4\pi \sigma a^2}{b} - 4\pi \sigma b \right) \]
Factoring out \( 4\pi \sigma \):
\[ V_B = \frac{4\pi \sigma}{b} \left( a^2 - b^2 \right) \]
Using \( k = \frac{1}{4\pi \epsilon_0} \):
\[ V_B = \frac{\sigma}{\epsilon_0} \left( \frac{a^2 - b^2}{b} \right) \]

### Step 4: Write the Potential of Shell C
Similarly, for shell C:
\[ V_C = k \left( \frac{Q_A}{c} + \frac{Q_B}{c} + \frac{Q_C}{c} \right) \]
Substituting the charges:
\[ V_C = k \left( \frac{\sigma \cdot 4\pi a^2}{c} - \frac{\sigma \cdot 4\pi b^2}{c} + \frac{\sigma \cdot 4\pi c^2}{c} \right) \]
Simplifying:
\[ V_C = k \left( \frac{4\pi \sigma a^2}{c} - \frac{4\pi \sigma b^2}{c} + 4\pi \sigma \right) \]
Factoring out \( 4\pi \sigma \):
\[ V_C = \frac{4\pi \sigma}{c} \left( a^2 - b^2 + c^2 \right) \]
Using \( k = \frac{1}{4\pi \epsilon_0} \):
\[ V_C = \frac{\sigma}{\epsilon_0} \left( \frac{a^2 - b^2 + c^2}{c} \right) \]

### Step 5: Compare with Given Expressions
We are given:
\[ V_B = \frac{\sigma}{\epsilon_0} \left( \frac{a^n}{b} - b + c \right) \]
\[ V_C = \frac{\sigma}{\epsilon_0} \left( \frac{a^n}{c} - \frac{b^n}{c} + c \right) \]

### Step 6: Identify \( n \)
From the expression for \( V_B \), we can see that:
\[ \frac{a^n}{b} - b + c \text{ must match } \frac{a^2 - b^2}{b} \]
This implies \( n = 2 \).

From the expression for \( V_C \):
\[ \frac{a^n}{c} - \frac{b^n}{c} + c \text{ must match } \frac{a^2 - b^2 + c^2}{c} \]
This also implies \( n = 2 \).

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