1+i^2+i^4+i^6+i^8+ ...+i^ 20 - Vidyadip

Question

$1+\text{i}^2+\text{i}^4+\text{i}^6+\text{i}^8+ ...+\text{i}^{20}$

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Answer

$1+\text{i}^2+\text{i}^4+\text{i}^6+\text{i}^8+ ...+\text{i}^{20}$
$=1+\text{i}^2+\text{i}^4+\text{i}^{4\times1}\times\text{i}^2+\text{i}^{4\times2}+\text{i}^{4\times2}\times\text{i}^{2}+\text{i}^{4\times3}+\text{i}^{4\times3}\times\text{i}^{2}+\text{i}^{4\times4}+\text{i}^{4\times4}\times\text{i}^2+\text{i}^{4\times5}$
$=1-1+1+1\times\text{i}^2+1+1\times\text{i}^2+1+1\times\text{i}^2+1+1\times\text{i}^2+1$
$=1-1+1-1+1-1+1-1+1-1+1$
$=1$

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