The sum of first n terms of an AP is given by `S_n=2n^2 + 3n`. Find the common difference of the AP.

Options

A. 3
B. 2
C. 6
D. 4

Explanation

To find the common difference of the arithmetic progression (AP) given that the sum of the first n terms \( S_n = 2n^2 + 3n \), we can follow these steps:

### Step 1: Understand the formula for the sum of the first n terms
The sum of the first n terms of an AP can be expressed as:
\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Calculate \( S_{n+1} \)
To find the common difference \( d \), we need to calculate \( S_{n+1} \):
\[ S_{n+1} = 2(n+1)^2 + 3(n+1) \]
Expanding this:
\[ S_{n+1} = 2(n^2 + 2n + 1) + 3(n + 1) = 2n^2 + 4n + 2 + 3n + 3 = 2n^2 + 7n + 5 \]

### Step 3: Find the common difference \( d \)
The common difference \( d \) can be found using the formula:
\[ d = S_{n+1} - S_n \]
Substituting the values we calculated:
\[ d = (2n^2 + 7n + 5) - (2n^2 + 3n) = 7n + 5 - 3n = 4n + 5 \]

### Step 4: Calculate \( d \) for a specific value of \( n \)
To find the common difference, we can substitute a specific value for \( n \). Let's use \( n = 1 \):
\[ d = 4(1) + 5 = 4 + 5 = 9 \]

### Final Answer
Thus, the common difference \( d \) of the AP is:
\[ \boxed{9} \]