The sum of15 terms of an A.P. is zero and its 4th term is 12. Find its 14th term.

Answer

To solve the problem step by step, we will follow the information given in the question and use the formulas related to Arithmetic Progression (A.P.).

### Step 1: Understanding the given information
We are given that:
- The sum of the first 15 terms of an A.P. is 0.
- The 4th term of the A.P. is 12.

### Step 2: Write the formula for the sum of the first n terms of an A.P.
The sum \( S_n \) of the first \( n \) terms of an A.P. can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]
where:
- \( A \) is the first term,
- \( D \) is the common difference,
- \( n \) is the number of terms.

### Step 3: Substitute the known values into the sum formula
For our case, \( n = 15 \) and \( S_{15} = 0 \):
\[ 0 = \frac{15}{2} \times (2A + (15 - 1)D) \]
This simplifies to:
\[ 0 = \frac{15}{2} \times (2A + 14D) \]
Since \( \frac{15}{2} \) is not zero, we can set the expression in parentheses to zero:
\[ 2A + 14D = 0 \]

### Step 4: Rearranging the equation
From the equation \( 2A + 14D = 0 \), we can express \( A \) in terms of \( D \):
\[ 2A = -14D \implies A = -7D \]

### Step 5: Use the information about the 4th term
The 4th term of an A.P. is given by:
\[ T_4 = A + (4 - 1)D = A + 3D \]
We know \( T_4 = 12 \):
\[ A + 3D = 12 \]

### Step 6: Substitute \( A \) in the equation
Substituting \( A = -7D \) into the equation \( A + 3D = 12 \):
\[ -7D + 3D = 12 \]
This simplifies to:
\[ -4D = 12 \]

### Step 7: Solve for \( D \)
Dividing both sides by -4:
\[ D = -3 \]

### Step 8: Find \( A \)
Now substitute \( D = -3 \) back into the equation \( A = -7D \):
\[ A = -7 \times (-3) = 21 \]

### Step 9: Find the 14th term
The 14th term \( T_{14} \) is given by:
\[ T_{14} = A + (14 - 1)D = A + 13D \]
Substituting the values of \( A \) and \( D \):
\[ T_{14} = 21 + 13 \times (-3) \]
Calculating this gives:
\[ T_{14} = 21 - 39 = -18 \]

### Final Answer
The 14th term of the A.P. is:
\[ \boxed{-18} \]