Show that

$1+i^{10}+i^{20}+i^{30}$

$=1+i^{4 \times 2+2}+i^{4 \times 5}+i^{4 \times 7+2}$

$=1+\left[\left(i^{4}\right)^{2} \times i^{2}\right]+\left(i^{4}\right)^{5}+\left[\left(i^{4}\right)^{7} \times i^{2}\right]$

$=1+i^{2}+1+i^{2}$         $\left(\because i^{4}=1\right)$

$=1-1+1-1$                       $\left(\because i^{2}=-1\right)$

= 0

This is a real number.

Hence proved.