[tex]\displaystyle\bf\\Trebuie~sa~aratam~ca\!:\\\\1+i+i^2+...+i^{10}=i\\\\1=i^0\\\\i=i^1\\\\\textbf{Scriem suma cu toti termenii pentru ca sunt putini.}\\\\i^0+i^1+i^2+i^3+i^4+i^5+i^6+i^7+i^8+i^9+i^{10}=[/tex]
.
[tex]\displaystyle\bf\\\textbf{Facem urmatoarea grupare a termenilor:}\\\\=(i^0+i^1+i^2+i^3)+(i^4+i^5+i^6+i^7)+(i^8+i^9+i^{10})=\\\\\textbf{Dam factor comun.}\\\\=(i^0+i^1+i^2+i^3)+i^4(i^0+i^1+i^2+i^3)+i^8(i^0+i^1+i^{2})\\\\\textbf{Calculam prima paranteza.}\\\\i^0=1\\i^1=i\\i^2=-1\\i^3=i^2\times i=-1\times i=-i\\\\\implies (i^0+i^1+i^2+i^3)=(1+i-1-i)=\boxed{\bf0}[/tex]
.
[tex]\displaystyle\bf\\\textbf{Calculam factorii comuni pe care i-am dat.}\\\\i^4=\Big(i^2\Big)^2=\Big(-1\Big)^2=\boxed{1}\\\\i^8=\Big(i^4\Big)^2=\Big(1\Big)^2=\boxed{1}\\\\\textbf{Calculam ultima paranteza}\\\\(i^0+i^1+i^{2}) = 1+i-1=\boxed{\bf~i~}\\\\\textbf{Calculam toata suma:}\\\\\underbrace{(i^0+i^1+i^2+i^3)}_{=0}+i^4\underbrace{(i^0+i^1+i^2+i^3)}_{=0}+i^8\underbrace{(i^0+i^1+i^{2})}_{=i}=\\\\=0+1\times0+1\times i=0+0+i=\boxed{\boxed{\bf~i~}}[/tex]
