If the sum of first p terms of an AP is `(ap^(2) +bp)`, find its common difference.

Explanation

To find the common difference of the arithmetic progression (AP) given that the sum of the first \( p \) terms is \( S_p = ap^2 + bp \), we can follow these steps:

### Step 1: Identify the formula for the sum of the first \( p \) terms of an AP
The sum of the first \( p \) terms of an arithmetic progression can be expressed as:
\[ S_p = \frac{p}{2} \times (2a + (p - 1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Set the two expressions for \( S_p \) equal
From the problem, we know:
\[ S_p = ap^2 + bp \]
Thus, we can set the two expressions for \( S_p \) equal to each other:
\[ \frac{p}{2} \times (2a + (p - 1)d) = ap^2 + bp \]

### Step 3: Multiply both sides by 2 to eliminate the fraction
Multiplying both sides by 2 gives:
\[ p \times (2a + (p - 1)d) = 2ap^2 + 2bp \]

### Step 4: Expand the left side
Expanding the left side results in:
\[ 2ap + p(p - 1)d = 2ap^2 + 2bp \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ p(p - 1)d = 2ap^2 + 2bp - 2ap \]
This simplifies to:
\[ p(p - 1)d = 2ap^2 + (2b - 2a)p \]

### Step 6: Factor out \( p \)
Since we are looking for the common difference \( d \), we can divide both sides by \( p \) (assuming \( p \neq 0 \)):
\[ (p - 1)d = 2ap + (2b - 2a) \]

### Step 7: Solve for \( d \)
Now, we can express \( d \) as:
\[ d = \frac{2ap + (2b - 2a)}{p - 1} \]

### Step 8: Find the common difference when \( p = 2 \)
To find the common difference, we can substitute \( p = 2 \):
\[ d = \frac{2a(2) + (2b - 2a)}{2 - 1} = \frac{4a + 2b - 2a}{1} = 2a + 2b \]

### Final Result
Thus, the common difference \( d \) is:
\[ d = 2a \]

---