Question Number 187947 by thean last updated on 24/Feb/23
Answered by witcher3 last updated on 24/Feb/23
$$\left(\mathrm{1}+\mathrm{i}\right)^{\mathrm{2000}} =\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{2000}} {\sum}}\mathrm{i}^{\mathrm{k}} \mathrm{C}_{\mathrm{2000}} ^{\mathrm{k}} =\left(\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{1000}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{k}} \mathrm{C}_{\mathrm{2000}} ^{\mathrm{2k}} +\mathrm{i}\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{999}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{k}} \mathrm{C}_{\mathrm{2000}} ^{\mathrm{2k}+\mathrm{1}} \right) \\ $$$$\mathrm{S}=\mathrm{Re}\left(\mathrm{1}+\mathrm{i}\right)^{\mathrm{2000}} =\mathrm{2}^{\mathrm{1000}} \mathrm{e}^{\mathrm{i500}\pi} =\left(\mathrm{2}\right)^{\mathrm{1000}} \\ $$