Question Number 207885 by hardmath last updated on 29/May/24
$$\mathrm{Find}: \\ $$$$\boldsymbol{\mathrm{i}}^{\mathrm{4}} \:+\:\boldsymbol{\mathrm{i}}^{\mathrm{8}} \:+\:\boldsymbol{\mathrm{i}}^{\mathrm{12}} \:+\:\boldsymbol{\mathrm{i}}^{\mathrm{16}} \:+\:\boldsymbol{\mathrm{i}}^{\mathrm{20}} \:+\:\boldsymbol{\mathrm{i}}^{\mathrm{24}} \:+...+\:\boldsymbol{\mathrm{i}}^{\mathrm{100}} \:=\:? \\ $$
Commented by mr W last updated on 29/May/24
$${do}\:{you}\:{mean}\:{i}=\sqrt{−\mathrm{1}}\:? \\ $$
Commented by hardmath last updated on 29/May/24
$$\mathrm{yes}\:\mathrm{professor}... \\ $$$$\mathrm{answer}:\:\mathrm{25} \\ $$
Commented by mr W last updated on 29/May/24
$${i}^{\mathrm{4}} ={i}^{\mathrm{8}} ={i}^{\mathrm{16}} =...={i}^{\mathrm{100}} =\mathrm{1} \\ $$$$\Sigma=\underset{\mathrm{25}\:{times}} {\mathrm{1}+\mathrm{1}+\mathrm{1}+...+\mathrm{1}}=\mathrm{25} \\ $$
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
$${i}=\sqrt{−\mathrm{1}} \\ $$$${i}^{\mathrm{2}} =−\mathrm{1} \\ $$$${i}^{\mathrm{4}} =\left(−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{1} \\ $$$${i}^{\mathrm{8}} ={i}^{\mathrm{4}} ×{i}^{\mathrm{4}} =\mathrm{1}×\mathrm{1}=\mathrm{1} \\ $$$$... \\ $$
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
$$\mathrm{i}^{\mathrm{4}{n}} =\mathrm{1} \\ $$$$\mathrm{i}^{\mathrm{4}{n}+\mathrm{1}} =\mathrm{i} \\ $$$$\mathrm{i}^{\mathrm{4}{n}+\mathrm{2}} =−\mathrm{1} \\ $$$$\mathrm{i}^{\mathrm{4}{n}+\mathrm{3}} =−\mathrm{i} \\ $$