If `S_(n)` the sum of first n terms of an A.P. is given by `Sn = 3n^(2) - 4n`, find the 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 given by the formula \( S_n = 3n^2 - 4n \), we can follow these steps:

### Step 1: Find the first term \( a_1 \)
The first term of the A.P. can be found by substituting \( n = 1 \) into the sum formula \( S_n \).

\[ S_1 = 3(1)^2 - 4(1) = 3 - 4 = -1 \]

Thus, \( a_1 = S_1 = -1 \).

### Step 2: Find the second term \( a_2 \)
The second term can be found by substituting \( n = 2 \) into the sum formula \( S_n \).

\[ S_2 = 3(2)^2 - 4(2) = 3(4) - 8 = 12 - 8 = 4 \]

Now, since \( S_2 = a_1 + a_2 \), we can find \( a_2 \):

\[ S_2 = a_1 + a_2 \implies 4 = -1 + a_2 \implies a_2 = 4 + 1 = 5 \]

### Step 3: Find the common difference \( d \)
The common difference \( d \) can be found by subtracting the first term from the second term:

\[ d = a_2 - a_1 = 5 - (-1) = 5 + 1 = 6 \]

### Step 4: Write the formula for the nth term
The formula for the nth term \( a_n \) of an A.P. is given by:

\[ a_n = a_1 + (n - 1) \cdot d \]

Substituting the values of \( a_1 \) and \( d \):

\[ a_n = -1 + (n - 1) \cdot 6 \]

### Step 5: Simplify the expression
Now, we simplify the expression:

\[ a_n = -1 + 6(n - 1) = -1 + 6n - 6 = 6n - 7 \]

### Final Answer
Thus, the nth term of the A.P. is:

\[ \boxed{6n - 7} \]

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