Write the 2nd term of the AP, if its `S_(n)=n^(2)+2n`.

Answer

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

### Step 1: Understand the formula for the nth term
The nth term \( A_n \) of an AP can be derived from the sum of the first n terms \( S_n \) using the formula:
\[ A_n = S_{n} - S_{n-1} \]
where \( S_{n-1} \) is the sum of the first \( n-1 \) terms.

### Step 2: Calculate \( S_{n-1} \)
To find \( S_{n-1} \), we replace \( n \) with \( n-1 \) in the expression for \( S_n \):
\[ S_{n-1} = (n-1)^2 + 2(n-1) \]
Expanding this:
\[ S_{n-1} = (n^2 - 2n + 1) + (2n - 2) = n^2 - 2n + 1 + 2n - 2 = n^2 - 1 \]

### Step 3: Calculate \( A_n \)
Now, we can find \( A_n \):
\[ A_n = S_n - S_{n-1} = (n^2 + 2n) - (n^2 - 1) \]
Simplifying this:
\[ A_n = n^2 + 2n - n^2 + 1 = 2n + 1 \]

### Step 4: Find the 2nd term \( A_2 \)
To find the second term, we substitute \( n = 2 \) into the expression for \( A_n \):
\[ A_2 = 2(2) + 1 = 4 + 1 = 5 \]

### Conclusion
The second term of the AP is \( 5 \).