If the 3rd and the 9th terms of an A.P. are 4 and -8 respectively, then which term of the A.P. is zero ?

We know that

    a_n = a + (n - 1)d

Given,

    a₃ = 4 and a₉ = -8
∴ a + 2d = 4 and a + 8d = -8

Subtracting the two Equations,

⇒ a + 8d - (a + 2d) = -8 - 4
⇒ a - a + 8d - 2d = -12
⇒ 6d = -12
⇒ d = -2.

Putting value of d in a + 2d = 4

⇒ a + 2 × (-2) = 4
⇒ a - 4 = 4
⇒ a = 4 + 4
⇒ a = 8.

Let nth term of the A.P. be zero so, a_n = 0.

⇒ a + (n - 1)d = 0
⇒ 8 + (n - 1) × (-2) = 0
⇒ 8 - 2n + 2 = 0
⇒ 10 - 2n = 0
⇒ 2n = 10
⇒ n = 5.

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