The sum of n terms of an A.P. is `(n^(2)+5n).` Its common difference is :

Options

A. 1
B. 4
C. 2
D. None of these

Explanation

To find the common difference of the arithmetic progression (A.P.) given that the sum of the first n terms is \( S_n = n^2 + 5n \), we can follow these steps:

### Step 1: Find the sum of the first term \( S_1 \)
The sum of the first term \( S_1 \) can be calculated by substituting \( n = 1 \) into the sum formula:
\[ S_1 = 1^2 + 5 \cdot 1 = 1 + 5 = 6 \]

### Step 2: Find the sum of the first two terms \( S_2 \)
Next, we calculate the sum of the first two terms \( S_2 \) by substituting \( n = 2 \):
\[ S_2 = 2^2 + 5 \cdot 2 = 4 + 10 = 14 \]

### Step 3: Relate the sums to the terms of the A.P.
In an A.P., the sum of the first n terms can be expressed as:
\[ S_n = a_1 + a_2 + a_3 + \ldots + a_n \]
Where \( a_1 \) is the first term and \( d \) is the common difference. We know:
- \( S_1 = a_1 \)
- \( S_2 = a_1 + a_2 \)

From our calculations:
- \( S_1 = 6 \) implies \( a_1 = 6 \)
- \( S_2 = 14 \) implies \( a_1 + a_2 = 14 \)

### Step 4: Solve for \( a_2 \)
Using the value of \( a_1 \):
\[ 6 + a_2 = 14 \]
Subtracting 6 from both sides gives:
\[ a_2 = 14 - 6 = 8 \]

### Step 5: Find the common difference \( d \)
The common difference \( d \) in an A.P. is given by:
\[ d = a_2 - a_1 \]
Substituting the values we found:
\[ d = 8 - 6 = 2 \]

### Conclusion
The common difference \( d \) of the A.P. is:
\[ \boxed{2} \]