Question:
Evaluate $\left(i^{57}+i^{70}+i^{91}+i^{101}+i^{104}\right)$
Solution:
We have, $\mathrm{i}^{57}+\mathrm{i}^{70}+\mathrm{i}^{91}+\mathrm{i}^{101}+\mathrm{i}^{104}$
$=\left(i^{4}\right)^{14} \cdot i+\left(i^{4}\right)^{17} \cdot i^{2}+\left(i^{4}\right)^{22} \cdot i^{3}+\left(i^{4}\right)^{25} \cdot i+\left(i^{4}\right)^{26}$
We know that, $i^{4}=1$
$\Rightarrow(1)^{14} \cdot i+(1)^{17} \cdot i^{2}+(1)^{22} \cdot i^{3}+(1)^{25} \cdot i+(1)^{26}$
$=i+i^{2}+i^{3}+i+1$
$=i-1-i+i+1$
$=i$