Question

Выполните действия:
1+$i^{5}$+$i^{10}$+...+$i^{100}$

Answer 1

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