Which term of the series ( 28−16𝑖)( 27−14𝑖)( 26−12𝑖) +⋯ is (i) purely real (ii) purely imaginary.

Clearly the given sequence forms an A.P. with 

first term (a) = 28 - 16i and common difference (d) = -1 + 2i

⇒ a_n = a + (n - 1)d = 28 - 16i + (n - 1)(-1 + 2i) = 28 - 16i - n + 2ni +1 - 2i = (29 - n) + 2(n - 9)i

(1)

For this term to be purely real its imaginary part should be equal to zero

⇒ 2(n - 9) = 0

⇒ n - 9 = 0

⇒ n = 9

Hence required term is 29 - 9 = 20

(2)

For this term to be purely imaginary its real part should be equal to zero

⇒ 29 - n = 0

⇒ n = 29

Hence required term is 2(29 - 9)i = 40i