We know that

i⁴ⁿ⁺¹ = i

i⁴ⁿ⁺² = i² = -1

i⁴ⁿ⁺³ = i³ = -i

i⁴ⁿ⁺⁴ = i⁴ = 1

i⁴ⁿ⁺¹ + i⁴ⁿ⁺² + i⁴ⁿ⁺³ + i⁴ⁿ⁺⁴ = i + (-1) + (-i) + 1

= 0

S = i + i² + i³ + …… upto 1000 terms

We can make the pair of 4 terms because we know that value is repeat after every 4^th terms. So, there are total 250 pairs are made and each pair have value equal to 0.

S = 0