The first and second term of a G.P. are `x^(-4) and x^(n)` respectively. If `x^(52)` is the `8^(th)` term, then find the value of n.

Explanation

To solve the problem, we need to find the value of \( n \) given the first term and second term of a geometric progression (G.P.) and the 8th term.

### Step-by-step Solution:

1. **Identify the Terms**:
- The first term \( a_1 = x^{-4} \)
- The second term \( a_2 = x^n \)
- The 8th term \( a_8 = x^{52} \)

2. **Find the Common Ratio**:
- The common ratio \( r \) of a G.P. can be calculated using the formula:
\[ r = \frac{a_2}{a_1} = \frac{x^n}{x^{-4}} = x^{n + 4} \]

3. **Use the Formula for the nth Term**:
- The formula for the nth term of a G.P. is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
- For the 8th term:
\[ a_8 = a_1 \cdot r^{7} \]
- Substituting the values we have:
\[ x^{52} = x^{-4} \cdot (x^{n + 4})^{7} \]

4. **Simplify the Equation**:
- Simplifying the right side:
\[ x^{52} = x^{-4} \cdot x^{7(n + 4)} \]
- This can be rewritten as:
\[ x^{52} = x^{-4 + 7(n + 4)} \]
- Combine the exponents:
\[ x^{52} = x^{7n + 28 - 4} = x^{7n + 24} \]

5. **Set the Exponents Equal**:
- Since the bases are the same, we can set the exponents equal to each other:
\[ 52 = 7n + 24 \]

6. **Solve for \( n \)**:
- Rearranging the equation:
\[ 7n = 52 - 24 \]
\[ 7n = 28 \]
\[ n = \frac{28}{7} = 4 \]

### Conclusion:
The value of \( n \) is \( 4 \).