In an arithmetic progression, if 6 is the third term, the ninth term…

The correct answer is Option (3) → Seventh

Step 1: Let the AP be defined as

$a, a+d, a+2d, \dots$

• a = first term
• d = common difference

Step 2: Use given information

Third term = 6

$a + 2d = 6 \quad \text{(Equation 1)}$

Ninth term exceeds seventh term by 3

$(a + 8d) - (a + 6d) = 3 ⟹2d = 3 ⟹d = \frac{3}{2} = 1.5$

Step 3: Find first term

$a + 2d = 6 ⟹ a + 2(1.5) = 6 ⟹ a + 3 = 6 ⟹ a = 3$

Step 4: Find which term is 12

$n\text{th term} = a + (n-1)d = 12$

$3 + (n-1)(1.5) = 12$

$(n-1) \cdot 1.5 = 9 ⟹ n-1 = 6 ⟹ n = 7$