Question:
Show that 1 + i10 + i20 + i30 is a real number.
Solution:
$1+i^{10}+i^{20}+i^{30}$
$=1+i^{4 \times 2+2}+i^{4 \times 5}+i^{4 \times 7+2}$
$=1+\left[\left(i^{4}\right)^{2} \times i^{2}\right]+\left(i^{4}\right)^{5}+\left[\left(i^{4}\right)^{7} \times i^{2}\right]$
$=1+i^{2}+1+i^{2}$ $\left(\because i^{4}=1\right)$
$=1-1+1-1$ $\left(\because i^{2}=-1\right)$
= 0
This is a real number.
Hence proved.