Simplify: $i^{2} + i^{3} + i^{4} + i^{5}$

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Solution

We know that $i^{2} = - 1$
$i^{3} = i^{2} \times i = - i$
$i^{4} = {(i^{2})}^{2} = {(- 1)}^{2} = 1$
$i^{5} = i^{4} \times i = 1 \times i = i$
$∴ i^{2} + i^{3} + i^{4} + i^{5} = - 1 - i + 1 + i = 0$

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