if n belongs to N,(i^5)+(i^6)+(i^7)+(i^8)+...upto (4n+3) terms

Question

it is from complex numbers of intermediate level

Answer 1

i⁴ⁿ=(i⁴)ⁿ=1ⁿ=1, for integer n∈ℕ, that is, integer n>0 if ℕ is the set of natural numbers.

i⁴ⁿ⁺¹=i, i⁴ⁿ⁺²=-1, i⁴ⁿ⁺³=-i.

i⁴ⁿ⁺¹+ i⁴ⁿ⁺²+i⁴ⁿ⁺³+i⁴ⁿ⁺⁴=i-1-i+1=0, since 4n+4=4(n+1) and n+1 is also an integer.

Therefore the sum of the series is 0.