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

Let first term of A.P. be a and common difference be d.

By formula,

a_n = a + (n - 1)d

Given,

3rd term = 4

⇒ a₃ = 4

⇒ a + (3 - 1)d = 4

⇒ a + 2d = 4 ……..(1)

Given,

9th term = -8

⇒ a₉ = -8

⇒ a + (9 - 1)d = -8

⇒ a + 8d = -8 ……..(2)

Subtracting equation (1) from (2), we get :

⇒ a + 8d - (a + 2d) = -8 - 4

⇒ a - a + 8d - 2d = -12

⇒ 6d = -12

⇒ d = $-\dfrac{12}{6}$ = -2.

Substituting value of d in equation (1), we get :

⇒ a + 2 × -2 = 4

⇒ a - 4 = 4

⇒ a = 4 + 4 = 8.

Let nth term of the A.P. be zero.

⇒ a_n = a + (n - 1)d

⇒ 0 = 8 + (n - 1) × (-2)

⇒ 0 = 8 - 2n + 2

⇒ 2n = 10

⇒ n = $\dfrac{10}{2}$ = 5.

Hence, 5th term of the A.P. is zero.