[tex]\displaystyle\bf\\Calculam~partea~reala~a~numarului:\\z=i^{20}+i^{21}+i^{22}\\\\i^2=-1\\\\i^4=i^{2\times2}=\Big(i^2\Big)^2=\Big(-1\Big)^2=1\\\\i^{20}=i^{4\times5}=\Big(i^4\Big)^5=\Big(1\Big)^5=1\\\\\implies~\boxed{\bf i^{20}=1}\\\\i^{21}=i^{20+1}=i^{20}\times i^1=1\times i=i\\\\\implies~\boxed{\bf i^{21}=i}\\\\i^{22}=i^{21+1}=i^{21}\times i^1=i\times i=-1\\\\\implies~\boxed{\bf i^{22}=-1}\\\\z=i^{20}+i^{21}+i^{22}=\\~z=1+i+(-1)=\\~z=1+i-1=\\~z=1-1+i=\\~z=0+i\\\\\implies~\boxed{\bf Re~z=0}[/tex]
