If `S_(n)=3n^(2)+2n`, then write `a_(2)`

Answer

To find the term \( a_2 \) from the given sum of the first \( n \) terms \( S_n = 3n^2 + 2n \), we can follow these steps:

### Step 1: Understand the relationship between \( S_n \) and \( a_n \)
The \( n \)-th term \( a_n \) can be found using the formula:
\[ a_n = S_n - S_{n-1} \]
where \( S_n \) is the sum of the first \( n \) terms and \( S_{n-1} \) is the sum of the first \( n-1 \) terms.

### Step 2: Calculate \( S_{n-1} \)
To find \( S_{n-1} \), substitute \( n-1 \) into the expression for \( S_n \):
\[ S_{n-1} = 3(n-1)^2 + 2(n-1) \]
Expanding this:
\[ S_{n-1} = 3(n^2 - 2n + 1) + 2(n - 1) \]
\[ = 3n^2 - 6n + 3 + 2n - 2 \]
\[ = 3n^2 - 4n + 1 \]

### Step 3: Find \( a_n \)
Now, we can find \( a_n \):
\[ a_n = S_n - S_{n-1} \]
Substituting the expressions we have:
\[ S_n = 3n^2 + 2n \]
\[ S_{n-1} = 3n^2 - 4n + 1 \]
So,
\[ a_n = (3n^2 + 2n) - (3n^2 - 4n + 1) \]
Simplifying this:
\[ = 3n^2 + 2n - 3n^2 + 4n - 1 \]
\[ = 6n - 1 \]

### Step 4: Calculate \( a_2 \)
Now, we can find \( a_2 \) by substituting \( n = 2 \):
\[ a_2 = 6(2) - 1 \]
\[ = 12 - 1 \]
\[ = 11 \]

### Final Answer
Thus, the required term \( a_2 \) is:
\[ \boxed{11} \]