Find the rth term of an A.P., sum of whose first n terms is `2n + 3n^(2)`.

Answer

To find the rth term of an arithmetic progression (A.P.) whose sum of the first n terms is given by \( S_n = 2n + 3n^2 \), we can follow these steps:

### Step 1: Find \( S_{n-1} \)
The sum of the first \( n-1 \) terms can be calculated by substituting \( n-1 \) into the sum formula:
\[ S_{n-1} = 2(n-1) + 3(n-1)^2 \]
Calculating this gives:
\[ S_{n-1} = 2n - 2 + 3(n^2 - 2n + 1) = 2n - 2 + 3n^2 - 6n + 3 \]
Combining like terms:
\[ S_{n-1} = 3n^2 - 4n + 1 \]

### Step 2: Find the rth term \( T_r \)
The rth term of the A.P. can be found using the formula:
\[ T_r = S_n - S_{n-1} \]
Substituting the expressions we have for \( S_n \) and \( S_{n-1} \):
\[ T_r = (2n + 3n^2) - (3n^2 - 4n + 1) \]
Distributing the negative sign:
\[ T_r = 2n + 3n^2 - 3n^2 + 4n - 1 \]
Combining like terms:
\[ T_r = 6n - 1 \]

### Final Answer
Thus, the rth term of the A.P. is:
\[ T_r = 6n - 1 \]

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