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The value of $i^{2 n}+i^{2 n+1}+i^{2 n+2}+i^{2 n+3}$, where $i$
$=\sqrt{-1}$, is
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$=\sqrt{-1}$, is
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$\begin{aligned} & \mathrm{i}^{2 \mathrm{n}}+\mathrm{i}^{2 \mathrm{n}}+1+\mathrm{i}^{2 \mathrm{n}+2}+\mathrm{i}^{2 \mathrm{n}} \cdot+{ }^{3} \\ &=\mathrm{i}^{2 \mathrm{n}}+\mathrm{i}^{2 \mathrm{n}} \cdot \mathrm{i}+\mathrm{i}^{2 \mathrm{n}} \cdot \mathrm{i}^{2}+{ }^{\mathrm{i} 2 \mathrm{n}} \cdot \mathrm{i}^{3} \\ &=\mathrm{i}^{2 \mathrm{n}}\left(1+\mathrm{i}+\mathrm{i}^{2}+\mathrm{i}^{3}\right)\left[\text { since }, \mathrm{i}^{2}=-1, \mathrm{i}^{3}=\mathrm{i}^{2} \mathrm{i}=-\mathrm{i}\right] \\ &=\mathrm{i}^{2 \mathrm{n}}(1+\mathrm{i}-1-\mathrm{i}) \\ &=\mathrm{i}^{2 \mathrm{n}}(0) \\ &=0 \end{aligned}$
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