If the sum of n terms of an A.P. is `2n^(2)+5n` then the `n^(th)` term will be

Options

A. `4n+3`
B. `4n+5`
C. `4n+6`
D. `4n+7`

Answer

To find the \( n^{th} \) term of the arithmetic progression (A.P.) given that the sum of the first \( n \) terms \( S_n \) is \( 2n^2 + 5n \), we can follow these steps:

### Step 1: Write the formula for the \( n^{th} \) term
The \( n^{th} \) term \( T_n \) of an A.P. can be expressed in terms of the sum of the first \( n \) terms as:
\[ T_n = S_n - S_{n-1} \]

### Step 2: Calculate \( S_{n-1} \)
We know that:
\[ S_n = 2n^2 + 5n \]
To find \( S_{n-1} \), we substitute \( n-1 \) into the sum formula:
\[ S_{n-1} = 2(n-1)^2 + 5(n-1) \]
Now, we simplify \( S_{n-1} \):
\[ S_{n-1} = 2(n^2 - 2n + 1) + 5(n - 1) \]
\[ = 2n^2 - 4n + 2 + 5n - 5 \]
\[ = 2n^2 + (5n - 4n) + (2 - 5) \]
\[ = 2n^2 + n - 3 \]

### Step 3: Substitute \( S_n \) and \( S_{n-1} \) into the formula for \( T_n \)
Now we can find \( T_n \):
\[ T_n = S_n - S_{n-1} \]
Substituting the values we found:
\[ T_n = (2n^2 + 5n) - (2n^2 + n - 3) \]

### Step 4: Simplify the expression for \( T_n \)
Now, we simplify:
\[ T_n = 2n^2 + 5n - 2n^2 - n + 3 \]
\[ = (2n^2 - 2n^2) + (5n - n) + 3 \]
\[ = 4n + 3 \]

### Final Answer
Thus, the \( n^{th} \) term \( T_n \) of the A.P. is:
\[ \boxed{4n + 3} \]