If sum of first n terms of an A.P is `2n^(2)` + 7n, write its nth term.

Answer

To find the nth term of the arithmetic progression (A.P.) given that the sum of the first n terms \( S_n \) is \( 2n^2 + 7n \), we can follow these steps:

### Step 1: Identify the formula for the nth term
The nth term \( a_n \) of an A.P. 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 \( S_{n-1} \)
Given \( S_n = 2n^2 + 7n \), we need to find \( S_{n-1} \). To do this, we substitute \( n-1 \) into the formula for \( S_n \):
\[ S_{n-1} = 2(n-1)^2 + 7(n-1) \]

### Step 3: Simplify \( S_{n-1} \)
Now, we simplify \( S_{n-1} \):
\[ S_{n-1} = 2(n^2 - 2n + 1) + 7(n - 1) \]
\[ = 2n^2 - 4n + 2 + 7n - 7 \]
\[ = 2n^2 + 3n - 5 \]

### Step 4: Find \( a_n \)
Now that we have both \( S_n \) and \( S_{n-1} \), we can find \( a_n \):
\[ a_n = S_n - S_{n-1} \]
\[ = (2n^2 + 7n) - (2n^2 + 3n - 5) \]
\[ = 2n^2 + 7n - 2n^2 - 3n + 5 \]
\[ = (7n - 3n) + 5 \]
\[ = 4n + 5 \]

### Final Answer
Thus, the nth term \( a_n \) of the A.P. is:
\[ \boxed{4n + 5} \]