Evaluate $1 + i^{2} + i^{4} + i^{6} + . . . + i^{2 n}$ .

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Solution

$1 + i^{2} + i^{4} + i^{6} + . . . + i^{2 n}$
If n is odd.
$\Rightarrow 1 - 1 + 1 - 1 + 1 - 1 = 0$
If n is even
$\Rightarrow 1 - 1 + 1 . . . + 1 = 1$

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