If `S_n` denotes the sum of first n terms of an arithmetic progression and an denotes the `n^(th)` term of the same A.P. given `S_n = n^2p` ; where `p,n in N`, then

Options

A. `a_(1) = p`
B. common difference `= 2p`
C. `S_(p) = p^(3)`
D. `a_(p) = 2p^(3) - p`

Answer

To solve the problem, we will analyze the given information step by step.

### Step 1: Understanding the Sum of the First n Terms
The sum of the first n terms of an arithmetic progression (AP) is given by the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Given Information
We are given that:
\[ S_n = n^2 p \]
This means:
\[ \frac{n}{2} \times (2a + (n-1)d) = n^2 p \]

### Step 3: Finding the First Term \( a \)
For \( n = 1 \):
\[ S_1 = a = 1^2 p = p \]
Thus, the first term \( a \) is:
\[ a = p \]

### Step 4: Finding \( S_2 \)
Now, let’s calculate \( S_2 \):
\[ S_2 = 2^2 p = 4p \]

### Step 5: Finding the Second Term \( a_2 \)
The second term \( a_2 \) can be calculated as:
\[ a_2 = S_2 - S_1 = 4p - p = 3p \]

### Step 6: Finding the Common Difference \( d \)
The common difference \( d \) is given by:
\[ d = a_2 - a_1 = 3p - p = 2p \]

### Step 7: Finding \( S_p \)
Next, we calculate \( S_p \):
\[ S_p = p^2 p = p^3 \]

### Step 8: Finding the p-th Term \( a_p \)
Using the formula for the sum of the first p terms:
\[ S_p = \frac{p}{2} \times (2a + (p-1)d) \]
Substituting \( S_p = p^3 \), \( a = p \), and \( d = 2p \):
\[ p^3 = \frac{p}{2} \times (2p + (p-1)(2p)) \]
This simplifies to:
\[ p^3 = \frac{p}{2} \times (2p + 2p^2 - 2p) \]
\[ p^3 = \frac{p}{2} \times 2p^2 \]
\[ p^3 = p^3 \]
This confirms that our calculations are consistent.

### Step 9: Conclusion
From the above calculations, we can conclude:
- \( a_1 = p \) (Option A is correct)
- \( d = 2p \) (Option B is correct)
- \( S_p = p^3 \) (Option C is correct)
- The formula for \( a_p \) does not match the given option D.

Thus, the correct options are A, B, and C.