Second term of the AP if the `S_(n) = n^(2) +n` is ………

Answer

To find the second term of the arithmetic progression (AP) given the sum of the first n terms \( S_n = n^2 + n \), we will follow these steps:

### Step-by-Step Solution:

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

2. **Find \( S_{n+1} \)**:
We need to calculate \( S_{n+1} \) using the given formula. We replace \( n \) with \( n+1 \):
\[ S_{n+1} = (n+1)^2 + (n+1) \]
Expanding this, we get:
\[ S_{n+1} = n^2 + 2n + 1 + n + 1 = n^2 + 3n + 2 \]

3. **Calculate \( a_n \)**:
The nth term \( a_n \) can be found using the formula:
\[ a_n = S_{n} - S_{n-1} \]
We already have \( S_n = n^2 + n \). Now, we need to find \( S_{n-1} \):
\[ S_{n-1} = (n-1)^2 + (n-1) = n^2 - 2n + 1 + n - 1 = n^2 - n \]
Now, substituting into the formula for \( a_n \):
\[ a_n = S_n - S_{n-1} = (n^2 + n) - (n^2 - n) = 2n \]

4. **Find the second term \( a_2 \)**:
To find the second term, we substitute \( n = 2 \) into \( a_n \):
\[ a_2 = 2 \cdot 2 = 4 \]

5. **Conclusion**:
Therefore, the second term of the AP is:
\[ \boxed{4} \]