The value of $1 + i^{2} + i^{4} + i^{6} + i^{8} + . . . . . . . . . . . . . . . . + i^{20}$ is :

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Solution

$1 + i^{2} + i^{4} + i^{6} + i^{8} + i^{10} + i^{12} + i^{14} + i^{16} + i^{18} + i^{20}$

$= 1 + (- 1) + 1 + (- 1) + 1 + (- 1) + 1 + (- 1) + 1 + (- 1) + 1$

$= 1$ (Ans)

Formula:
$i^{2} = - 1$ and $i^{4} = 1$

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Q. Find the value of

1 + i^{2} + i^{4} + i^{6} + i^{8} + . . . . . + i^{20}

Find the values of the following expressions :

$(i) i^{49} + i^{68} + i^{89} + i^{110} (i i) i^{30} + i^{80} + i^{120} (i i i) i + i^{2} + i^{3} + i^{4} (i v) i^{5} + i^{10} + i^{15} (v) \frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} (v i) 1 + i^{2} + i^{4} + i^{6} + i^{8} + . . . + i^{20}$