If the sum of n terms of an AP is given by ` S_(n) = (2n^(2)+3n)` then find its common differnece.

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: Find \( S_{n-1} \)
We need to calculate \( S_{n-1} \) using the formula for \( S_n \):
\[ S_{n-1} = 2(n-1)^2 + 3(n-1) \]
Expanding this:
\[ = 2(n^2 - 2n + 1) + 3(n - 1) \]
\[ = 2n^2 - 4n + 2 + 3n - 3 \]
\[ = 2n^2 - n - 1 \]

### Step 2: Find the nth term \( t_n \)
The nth term \( t_n \) of the AP can be found using the relationship:
\[ t_n = S_n - S_{n-1} \]
Substituting the values we have:
\[ t_n = (2n^2 + 3n) - (2n^2 - n - 1) \]
\[ = 2n^2 + 3n - 2n^2 + n + 1 \]
\[ = 4n + 1 \]

### Step 3: Find the common difference \( d \)
The common difference \( d \) can be calculated as:
\[ d = t_n - t_{n-1} \]
First, we need to find \( t_{n-1} \):
\[ t_{n-1} = 4(n-1) + 1 = 4n - 4 + 1 = 4n - 3 \]
Now, substituting back to find \( d \):
\[ d = t_n - t_{n-1} = (4n + 1) - (4n - 3) \]
\[ = 4n + 1 - 4n + 3 \]
\[ = 4 \]

### Final Answer
The common difference \( d \) of the arithmetic progression is \( 4 \).

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