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Question Number 207885 by hardmath last updated on 29/May/24

$$\mathrm{Find}:$$$${\mathbf{i}}^4+{\mathbf{i}}^8+{\mathbf{i}}^{12}+{\mathbf{i}}^{16}+{\mathbf{i}}^{20}+{\mathbf{i}}^{24}+...+{\mathbf{i}}^{100}=?$$

Commented by mr W last updated on 29/May/24

$$doyoumeani=\sqrt{-1}?$$

Commented by hardmath last updated on 29/May/24

$$\mathrm{yes}\mathrm{professor}...$$$$\mathrm{answer}:25$$

Commented by mr W last updated on 29/May/24

$$i^4=i^8=i^{16}=...=i^{100}=1$$$$\mathrm{\Sigma }=\underset{25times}{1+1+1+...+1}=25$$

Commented by hardmath last updated on 29/May/24

$$$$Professor, is there a golden rule for these types of examples?

Commented by mr W last updated on 29/May/24

$$i=\sqrt{-1}$$$$i^2=-1$$$$i^4={(-1)}^2=1$$$$i^8=i^4\times i^4=1\times 1=1$$$$...$$

Commented by hardmath last updated on 29/May/24

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}$$

Commented by Frix last updated on 29/May/24

$${\mathrm{i}}^{4n}=1$$$${\mathrm{i}}^{4n+1}=\mathrm{i}$$$${\mathrm{i}}^{4n+2}=-1$$$${\mathrm{i}}^{4n+3}=-\mathrm{i}$$

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