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

We know that,

n^th term of an A.P. is given by,

a_n = a + (n - 1)d

Given, 3^rd term is 4

∴ a₃ = a + (3 - 1)d

⇒ 4 = a + 2d

⇒ a + 2d = 4 ……..(i)

Given, 9^th term is -8

∴ a₉ = a + (9 - 1)d

⇒ -8 = a + 8d

⇒ a + 8d = -8 ……..(ii)

Subtracting (i) from (ii) we get,

⇒ a + 8d - (a + 2d) = -8 - 4

⇒ 6d = -12

⇒ d = -2.

Substituting value of d in (i) we get,

⇒ a + 2(-2) = 4

⇒ a - 4 = 4

⇒ a = 8.

Let n^th term be zero.

∴ a_n = a + (n - 1)d = 0

⇒ 8 + (n - 1)(-2) = 0

⇒ 8 - 2n + 2 = 0

⇒ 2n = 10

⇒ n = 5.

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