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.) where the sum of the first n terms is given by \( S_n = n^2 + 5n \), we can follow these steps:

### Step 1: Find \( S_1 \)
The first step is to calculate the sum of the first term \( S_1 \) by substituting \( n = 1 \) into the sum formula.

\[ S_1 = 1^2 + 5 \cdot 1 = 1 + 5 = 6 \]

This means that the first term \( a_1 \) of the A.P. is:

\[ a_1 = S_1 = 6 \]

### Step 2: Find \( S_2 \)
Next, we calculate the sum of the first two terms \( S_2 \) by substituting \( n = 2 \) into the sum formula.

\[ S_2 = 2^2 + 5 \cdot 2 = 4 + 10 = 14 \]

This means that the sum of the first two terms \( a_1 + a_2 \) is:

\[ S_2 = a_1 + a_2 = 14 \]

### Step 3: Solve for \( a_2 \)
Now we can substitute the value of \( a_1 \) into the equation for \( S_2 \):

\[ 14 = 6 + a_2 \]

To find \( a_2 \), we rearrange the equation:

\[ a_2 = 14 - 6 = 8 \]

### Step 4: Find the common difference \( d \)
The common difference \( d \) of the A.P. can be found using the formula:

\[ d = a_2 - a_1 \]

Substituting the values we found:

\[ d = 8 - 6 = 2 \]

### Conclusion
Thus, the common difference of the A.P. is:

\[ \boxed{2} \]

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