If the 3rd and the 9th terms of an arithmetic progression are 4 and -8 respectively, which term of it is zero ?

Answer

To solve the problem, we need to find which term of the arithmetic progression (AP) is zero, given that the 3rd term is 4 and the 9th term is -8. Let's go through the solution step by step.

### Step 1: Define the terms of the AP
The n-th term of an arithmetic progression can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Write equations for the given terms
From the problem, we know:
- The 3rd term \( T_3 = 4 \)
- The 9th term \( T_9 = -8 \)

Using the formula for the n-th term, we can write:
1. For the 3rd term:
\[ T_3 = a + (3 - 1) \cdot d = a + 2d = 4 \] (Equation 1)

2. For the 9th term:
\[ T_9 = a + (9 - 1) \cdot d = a + 8d = -8 \] (Equation 2)

### Step 3: Set up the equations
Now we have two equations:
1. \( a + 2d = 4 \) (Equation 1)
2. \( a + 8d = -8 \) (Equation 2)

### Step 4: Eliminate \( a \)
To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 8d) - (a + 2d) = -8 - 4 \]
This simplifies to:
\[ 6d = -12 \]
Dividing both sides by 6 gives:
\[ d = -2 \]

### 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
Now we know \( a = 8 \) and \( d = -2 \). We need to find \( n \) such that:
\[ T_n = 0 \]
Using the formula for the n-th term:
\[ 0 = a + (n - 1) \cdot d \]
Substituting the values of \( a \) and \( d \):
\[ 0 = 8 + (n - 1)(-2) \]
This simplifies to:
\[ 0 = 8 - 2(n - 1) \]
Rearranging gives:
\[ 2(n - 1) = 8 \]
Dividing both sides by 2:
\[ n - 1 = 4 \]
Adding 1 to both sides gives:
\[ n = 5 \]

### Conclusion
Thus, the term of the arithmetic progression that is zero is the **5th term**.

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