for n in the natural numbers prove that (2+i)^n is never real.

base case

(2+i) is not real

inductive hypothesis

for k=n assume (2+i)^k is not in the reals

must prove

(2+i)^(k+1)

Ive got to the point where

2(2+i)^k + i(2+i)^k must not be real. its seems logical that the i terms would not cancel out making this sum not real but i cant figure out a way to prove it.

I've tried using the binomial theorem. Any HINTS would be greatly appreciated.