we have, 1 + i¹⁰ + i²⁰ + i³⁰

= 1 + (i⁴)².i² + (i⁴)⁵ + (i⁴)⁷.i²

We know that, i⁴ = 1

⇒ 1 + (1)².i² + (1)⁵ + (1)⁷.i²

= 1 + i² + 1 + i²

= 1 -1 + 1 -1

= 0