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Question Number 187947 by thean last updated on 24/Feb/23

Answered by witcher3 last updated on 24/Feb/23

$${(1+\mathrm{i})}^{2000}=\underset{\mathrm{k}=0}{\sum ^{2000}}{\mathrm{i}}^{\mathrm{k}}{\mathrm{C}}_{2000}^{\mathrm{k}}=(\underset{\mathrm{k}=0}{\sum ^{1000}}{(-1)}^{\mathrm{k}}{\mathrm{C}}_{2000}^{2\mathrm{k}}+\mathrm{i}\underset{\mathrm{k}=0}{\sum ^{999}}{(-1)}^{\mathrm{k}}{\mathrm{C}}_{2000}^{2\mathrm{k}+1})$$$$\mathrm{S}=\mathrm{Re}{(1+\mathrm{i})}^{2000}=2^{1000}{\mathrm{e}}^{\mathrm{i}500\pi }={\left(2\right)}^{1000}$$

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