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

Options

A. A. `4n-3`
B. B. `3n-4`
C. C. `4n+3`
D. D. `3n +9`

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 use the formula for the \( n^{th} \) term of an A.P., which is:

\[ T_n = S_n - S_{n-1} \]

### Step 1: Calculate \( S_{n-1} \)

First, we need to find \( S_{n-1} \). We can substitute \( n-1 \) into the expression for \( S_n \):

\[ S_{n-1} = 2(n-1)^2 + 5(n-1) \]

Expanding this:

\[ S_{n-1} = 2(n^2 - 2n + 1) + 5(n - 1) \]
\[ = 2n^2 - 4n + 2 + 5n - 5 \]
\[ = 2n^2 + (5n - 4n) + (2 - 5) \]
\[ = 2n^2 + n - 3 \]

### Step 2: Calculate \( T_n \)

Now we can find \( T_n \):

\[ T_n = S_n - S_{n-1} \]
\[ = (2n^2 + 5n) - (2n^2 + n - 3) \]

Distributing the negative sign:

\[ = 2n^2 + 5n - 2n^2 - n + 3 \]

Now, simplifying:

\[ = (2n^2 - 2n^2) + (5n - n) + 3 \]
\[ = 0 + 4n + 3 \]
\[ = 4n + 3 \]

### Final Answer

Thus, the \( n^{th} \) term \( T_n \) of the A.P. is:

\[ \boxed{4n + 3} \]