If i^n+2 = -i then what could be the value of n

Answer 1

iⁿ + 2 = -i ==> iⁿ + i = -2. Let's take the square of both sides ==> (i²)ⁿ + i² + 2 x i x iⁿ = 4

(-1)n - 1 + 2 x iⁿ⁺¹ = 4

-n - 1 - 4 = 2 x iⁿ⁺¹

-(n+5)÷2 = i^n+1. (The only two options for the right side is either 1 or -1 as the left hand side has no "i" fully real.

If you take, 1, -(n+5) = 2, n = -7 ( Plug this to the right side and see that i^-6 = (i²)^-3= -1 ( NOT CONFIRMED )

If yi take -1, then n = -3, ( Plug this to the right side and see that i^-2 = -1 (NOT CONFIRMED)

Dear Khadija;

I am not 100% sure, but it seems there is no such 'n' value to satisy this. Please review critically and let me know if this could be improved

Best

Bilal

Answer 2

i² = -1 and i³ = -i

Therefore n+2=3 works.