If the sum of n terms of an A.P. is `(pn+qn^(2)),` where p and q are constants, find the common difference.

Explanation

To find the common difference of the arithmetic progression (A.P.) given that the sum of the first n terms \( S_n \) is expressed as \( S_n = pn + qn^2 \), where \( p \) and \( q \) are constants, we can follow these steps:

### Step 1: Find the first term \( S_1 \)
The sum of the first term \( S_1 \) can be calculated by substituting \( n = 1 \) into the equation for \( S_n \):
\[ S_1 = p(1) + q(1^2) = p + q \]

### Step 2: Find the second term \( S_2 \)
Next, we calculate the sum of the first two terms \( S_2 \) by substituting \( n = 2 \):
\[ S_2 = p(2) + q(2^2) = 2p + 4q \]

### Step 3: Find the first term \( a_1 \)
The first term of the A.P. is equal to \( S_1 \):
\[ a_1 = S_1 = p + q \]

### Step 4: Find the second term \( a_2 \)
The second term of the A.P. can be found using the formula \( a_2 = S_2 - S_1 \):
\[ a_2 = S_2 - S_1 = (2p + 4q) - (p + q) = 2p + 4q - p - q = p + 3q \]

### Step 5: Calculate the common difference \( d \)
The common difference \( d \) of the A.P. is given by the difference between the second term and the first term:
\[ d = a_2 - a_1 = (p + 3q) - (p + q) = p + 3q - p - q = 2q \]

### Conclusion
Thus, the common difference \( d \) is:
\[ d = 2q \]