Выполните действия: 1+[latex] i^{5} [/latex]+[latex] i^{10} [/latex]+...+[latex] i^{100} [/latex]

[latex]i^{5}=(i^2)^2\cdot i=(-1)^2\cdot i=i\\\\i^{10}=(i^5)^2=i^2=-1\\\\i^{15}=i^{10}\cdot i^5=-1\cdot i=-i\\\\i^{20}=(i^{10})^2=(-1)^2=1\\\\i^{25}=i^{20}\cdot i^5=1\cdot i=i\\\\i^{30}=(i^{10})^3=(-1)^3=-1\\\\.........................................\\\\\i^{90}=-1\\\\i^{95}=-i\\\\i^{100}=1\\\\1+i^5+i^{10}+i^{15}+...+i^{90}+i^{95}+i^{100}=\\\\=1+i+1-i+...-1-i+1=1+i-i=1[/latex]