The third and ninth terms of an arithmetic progression are 4 and -8 respectively. Which term of this will be zero?

Let first term be a and common difference be d.
We know that, a_n = a + (n - 1)d

Where,

a = First term of AP
d = Common difference of AP
and no of terms is ‘n’

∴ a₃ = a + (3 - 1)d
⇒ 4 = a + 2d …(i)
and a₉ = a + (9 - 1)d
⇒ -8 = a + 8d …(ii)

On solving eq. (i) and (ii), we get,
a = 8 and d = -2

Let the n^th term be zero.
⇒ a_n = a + (n - 1)d
⇒ 0 = 8 + (n - 1) × (-2)
⇒ 0 = 8 – 2n -2(-1)
⇒ 0 = 8 – 2n + 2
⇒ -10 = – 2n
⇒ n = 5

Hence, 5^th will be zero.