Question

Show that $1+i^{10}+i^{20}+i^{30}$ is a real number.

Answer 1

$1+i^{10}+i^{20}+i^{30}=1+i^{4\times 2}+i^2+i^{4\times 5}+i^{4\times 7}\times i^2=1-1+1-1=0,\text{which is real number.}$