If the third and the 9th terms of an A.P. be 4 and -8 respectively, find which term is zero ?

Answer

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

### Step 1: Identify the given terms
We know that:
- The third term \( A_3 = 4 \)
- The ninth term \( A_9 = -8 \)

### Step 2: Write the formula for the nth term of an A.P.
The nth term of an A.P. can be expressed as:
\[ A_n = A + (n - 1)D \]
where \( A \) is the first term and \( D \) is the common difference.

### Step 3: Set up equations for the given terms
From the information provided:
1. For the third term:
\[ A_3 = A + (3 - 1)D = A + 2D = 4 \quad \text{(Equation 1)} \]
2. For the ninth term:
\[ A_9 = A + (9 - 1)D = A + 8D = -8 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( A + 2D = 4 \) (Equation 1)
2. \( A + 8D = -8 \) (Equation 2)

To eliminate \( A \), we can subtract Equation 1 from Equation 2:
\[ (A + 8D) - (A + 2D) = -8 - 4 \]
This simplifies to:
\[ 6D = -12 \]
Now, divide both sides by 6:
\[ D = -2 \]

### Step 5: Substitute \( D \) back to find \( A \)
Now that we have \( D \), we can substitute it back into Equation 1 to find \( A \):
\[ A + 2(-2) = 4 \]
This simplifies to:
\[ A - 4 = 4 \]
Adding 4 to both sides gives:
\[ A = 8 \]

### Step 6: Find which term is zero
We need to find \( n \) such that \( A_n = 0 \):
\[ A_n = A + (n - 1)D = 0 \]
Substituting the values of \( A \) and \( D \):
\[ 8 + (n - 1)(-2) = 0 \]
This simplifies to:
\[ 8 - 2(n - 1) = 0 \]
Rearranging gives:
\[ 8 = 2(n - 1) \]
Dividing both sides by 2:
\[ 4 = n - 1 \]
Adding 1 to both sides results in:
\[ n = 5 \]

### Final Answer
Thus, the fifth term of the A.P. is 0.

---