Question

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

Hint:

To find the nth term of an AP, use the formula \(T_n = a + (n - 1) \cdot d\), and solve for the required values.

Answer 1

Step 1: General form of the nth term of an AP.
The nth term of an arithmetic progression (AP) is given by: \[ T_n = a + (n - 1) \cdot d \] where \(a\) is the first term and \(d\) is the common difference.
Step 2: Use the given terms to find \(a\) and \(d\).
We are given that the 3rd term \(T_3 = 4\) and the 9th term \(T_9 = -8\). For the 3rd term: \[ T_3 = a + (3 - 1) \cdot d = a + 2d = 4 \quad \text{(Equation 1)} \] For the 9th term: \[ T_9 = a + (9 - 1) \cdot d = a + 8d = -8 \quad \text{(Equation 2)} \] Now solve these two equations simultaneously. Subtract Equation 1 from Equation 2: \[ (a + 8d) - (a + 2d) = -8 - 4 \] \[ 6d = -12 \quad \Rightarrow \quad d = -2 \] Substitute \(d = -2\) into Equation 1: \[ a + 2(-2) = 4 \quad \Rightarrow \quad a - 4 = 4 \quad \Rightarrow \quad a = 8 \]
Step 3: Find the term that is zero.
We want to find the term where \(T_n = 0\): \[ T_n = a + (n - 1) \cdot d = 8 + (n - 1) \cdot (-2) = 0 \] \[ 8 - 2(n - 1) = 0 \quad \Rightarrow \quad 8 - 2n + 2 = 0 \quad \Rightarrow \quad 10 - 2n = 0 \quad \Rightarrow \quad n = 5 \] Thus, the 5th term of the AP is zero.