Question:
Evaluate $\left(1+i^{10}+i^{20}+i^{30}\right)$
Solution:
We have, $1+\mathrm{i}^{10}+\mathrm{i}^{20}+\mathrm{i}^{30}$
$=1+\left(i^{4}\right)^{2} \cdot i^{2}+\left(i^{4}\right)^{5}+\left(i^{4}\right)^{7} \cdot i^{2}$
We know that, $i^{4}=1$
$\Rightarrow 1+(1)^{2} \cdot i^{2}+(1)^{5}+(1)^{7} \cdot i^{2}$
$=1+i^{2}+1+i^{2}$
$=1-1+1-1$
$=0$