If the sum of first n terms of an A.P. is `3n^(2)-2n`, then its 19th term is

Options

A. 123
B. 118
C. 109
D. 107

Answer

To find the 19th term of the given arithmetic progression (A.P.) where the sum of the first n terms is given by \( S_n = 3n^2 - 2n \), we can follow these steps:

### Step 1: Understand the formula for the nth term
The nth term \( a_n \) of an A.P. can be calculated 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 \( S_{n-1} \)
We need to find \( S_{n-1} \). Since \( S_n = 3n^2 - 2n \), we can substitute \( n-1 \) into the formula:
\[ S_{n-1} = 3(n-1)^2 - 2(n-1) \]
Expanding this:
\[ S_{n-1} = 3(n^2 - 2n + 1) - 2(n - 1) \]
\[ = 3n^2 - 6n + 3 - 2n + 2 \]
\[ = 3n^2 - 8n + 5 \]

### Step 3: Calculate the nth term \( a_n \)
Now we can find \( a_n \):
\[ a_n = S_n - S_{n-1} \]
Substituting the values:
\[ a_n = (3n^2 - 2n) - (3n^2 - 8n + 5) \]
Distributing the negative sign:
\[ = 3n^2 - 2n - 3n^2 + 8n - 5 \]
The \( 3n^2 \) terms cancel out:
\[ = (-2n + 8n) - 5 \]
\[ = 6n - 5 \]

### Step 4: Find the 19th term
To find the 19th term, substitute \( n = 19 \) into the formula for \( a_n \):
\[ a_{19} = 6(19) - 5 \]
Calculating this:
\[ = 114 - 5 \]
\[ = 109 \]

### Final Answer
Thus, the 19th term of the arithmetic progression is \( \boxed{109} \).