Question Number 185165 by aba last updated on 17/Jan/23
$$\mathrm{Simplify}\::\:\frac{\mathrm{i}^{\mathrm{2022}} +\mathrm{i}^{\mathrm{2023}} +\mathrm{i}^{\mathrm{2024}} +\mathrm{i}^{\mathrm{2025}} }{\mathrm{1}+\mathrm{i}} \\ $$
Answered by HeferH last updated on 18/Jan/23
$$\:{if}\::\:{i}\:=\:\sqrt{−\mathrm{1}} \\ $$$$\:{i}^{\mathrm{2}} =\:−\mathrm{1} \\ $$$$\:{i}^{\mathrm{3}} =\:−{i}\: \\ $$$$\:\:\Rightarrow\: \\ $$$$\:\frac{{i}^{\mathrm{2022}} \left(\mathrm{1}+{i}+{i}^{\mathrm{2}} +{i}^{\mathrm{3}} \right)}{\mathrm{1}+{i}}\:=\:\frac{{i}^{\mathrm{2022}} \left(\mathrm{1}+{i}\:−\mathrm{1}\:−{i}\right)}{\mathrm{1}\:+\:{i}}=\:\mathrm{0} \\ $$