Sum of first terms of an `A.P.` is 9 less than the sum of next 3 terms of the same `A.P.` and second term is 2 then `4^(th)` term of that `A.P.` is :

Options

A. `12`
B. `4`
C. `15`
D. `18`

Explanation

To solve the problem step-by-step, we will use the properties of an Arithmetic Progression (A.P.).

### Step 1: Understand the given information
We know that:
- The sum of the first three terms of an A.P. is 9 less than the sum of the next three terms.
- The second term of the A.P. is 2.

Let the first term of the A.P. be \( a \) and the common difference be \( d \).

### Step 2: Write the expressions for the sums
The first three terms of the A.P. are:
- \( a \), \( a + d \), \( a + 2d \)

The sum of the first three terms, \( S_3 \), is:
\[ S_3 = a + (a + d) + (a + 2d) = 3a + 3d \]

The next three terms of the A.P. are:
- \( a + 3d \), \( a + 4d \), \( a + 5d \)

The sum of the next three terms, \( S_6 - S_3 \), is:
\[ S_6 = a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) + (a + 5d) = 6a + 15d \]
Thus, the sum of the next three terms is:
\[ S_6 - S_3 = (6a + 15d) - (3a + 3d) = 3a + 12d \]

### Step 3: Set up the equation based on the problem statement
According to the problem, we have:
\[ S_3 + 9 = S_6 - S_3 \]
Substituting the expressions we derived:
\[ 3a + 3d + 9 = 3a + 12d \]

### Step 4: Simplify the equation
Subtract \( 3a \) from both sides:
\[ 3d + 9 = 12d \]
Now, subtract \( 3d \) from both sides:
\[ 9 = 9d \]
Dividing both sides by 9 gives:
\[ d = 1 \]

### Step 5: Use the second term information
We know the second term \( A_2 \) is given as:
\[ A_2 = a + d = 2 \]
Substituting \( d = 1 \):
\[ a + 1 = 2 \]
Thus, solving for \( a \):
\[ a = 1 \]

### Step 6: Find the fourth term
The fourth term \( A_4 \) is given by:
\[ A_4 = a + 3d \]
Substituting the values of \( a \) and \( d \):
\[ A_4 = 1 + 3 \cdot 1 = 1 + 3 = 4 \]

### Final Answer
The fourth term of the A.P. is \( \boxed{4} \).