Resposta:

[tex]\textsf{Leia abaixo}[/tex]

Explicação passo a passo:

[tex]\mathsf{S = i + i^2 + i^3 + i^4 + ... + i^{100}}[/tex]

[tex]\mathsf{i^1 = i}[/tex]

[tex]\mathsf{i^{100} = (i^2)^{50} = (-1)^{50} = 1}[/tex]

[tex]\mathsf{i^2 = -1}[/tex]

[tex]\mathsf{i^{99} = i.[\:(i^2)\:]^{49} = i.(-1)^{49} = -i}[/tex]

[tex]\mathsf{i^3 = i.i^2 = i.(-1) = -i}[/tex]

[tex]\mathsf{i^{98} = [\:(i^2)\:]^{49} = (-1)^{49} = -1}[/tex]

[tex]\mathsf{i^4 = (i^2)^2 = (-1)^2 = 1}[/tex]

[tex]\mathsf{i^{97} = i.[\:(i^2)\:]^{48} = i.(-1)^{48} = i}[/tex]

[tex]\boxed{\boxed{\mathsf{S = i + i^2 + i^3 + i^4 + ... + i^{100} = 0}}}[/tex]