Answer:

Given,

AP of 12 + 8i, 11 + 6i, 10 + 4i, …

a₁ = a = 12 + 8i

a₂ = 11 + 6i

Common difference,

d = a₂ – a₁

= 11 + 6i – (12 + 8i)

= 11 – 12 + 6i – 8i

= -1 – 2i

a_n = a + (n – 1) d

a_n = 12 + 8i + (n – 1) -1 – 2i

= 12 + 8i – n – 2ni + 1 + 2i

= 13 + 10i – n – 2ni

= (13 – n) + (10 – 2n) i

10 – 2n = 0

2n = 10

n = 10/2

= 5

5^th term is purely real.

∴ 13 – n = 0

n = 13

13^th term is purely imaginary.