Question:
Prove that $\left(1+i^{10}+i^{20}+i^{30}\right)$ is a real number.
Solution:
L.H.S $=\left(1+\mathrm{i}^{10}+\mathrm{i}^{20}+\mathrm{j}^{30}\right)$
$=\left(1+i^{4 \times 2+2}+i^{4 \times 5}+i^{4 \times 7+2}\right)$
Since $\Rightarrow i^{4 n}=1$
$\Rightarrow i^{4 n+1}=i$
$\Rightarrow i^{4 n+2}=-1$
$\Rightarrow i^{4 n+3}=-1$
$=1+i^{2}+1+i^{2}$
$=1+-1+1+-1$
= 0, which is a real no
Hence, $\left(1+\mathrm{i}^{10}+\mathrm{i}^{20}+\mathrm{i}^{30}\right)$ is a real number.