If `S_(n) = n^(2) p and S_(m) = m^(2) p, m ne n`, in A.P., then `S_(p)` is

Options

A. `p^(2)`
B. `p^(3)`
C. `p^(4)`
D. none

Answer

To solve the problem, we need to find the value of \( S_p \) given that \( S_n = n^2 p \) and \( S_m = m^2 p \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the A.P. Condition**:
Since \( S_n \) and \( S_m \) are in A.P., we can use the property of A.P. which states that the middle term is the average of the other two terms. Thus, we have:
\[ S_m = \frac{S_n + S_p}{2} \]

2. **Substituting the Given Values**:
We know that \( S_n = n^2 p \) and \( S_m = m^2 p \). Substituting these into the A.P. condition gives:
\[ m^2 p = \frac{n^2 p + S_p}{2} \]

3. **Clearing the Fraction**:
To eliminate the fraction, multiply both sides by 2:
\[ 2m^2 p = n^2 p + S_p \]

4. **Rearranging the Equation**:
Now, isolate \( S_p \):
\[ S_p = 2m^2 p - n^2 p \]

5. **Factoring Out \( p \)**:
Factor \( p \) out of the right-hand side:
\[ S_p = p(2m^2 - n^2) \]

6. **Using the Difference of Squares**:
The expression \( 2m^2 - n^2 \) can be rewritten as:
\[ S_p = p((\sqrt{2}m)^2 - n^2) = p(\sqrt{2}m - n)(\sqrt{2}m + n) \]

7. **Final Expression**:
Thus, we have derived the expression for \( S_p \):
\[ S_p = p(2m^2 - n^2) \]

### Final Answer:
\[ S_p = p(2m^2 - n^2) \]