If n is any positive integer, write the value of i^ 4n+1 -i^ 4n-1…

Question

If n is any positive integer, write the value of $\frac{\text{i}^{4\text{n}+1}-\text{i}^{4\text{n}-1}}{2}.$

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Answer

$\frac{\text{i}^{4\text{n}+1}-\text{i}^{4\text{n}-1}}{2}$
$=\frac{\text{i}-\frac{1}{\text{i}}}{2} \ \big(\because\text{i}^{4\text{n}}=1,\text{i}^{-1}=\frac{1}{\text{i}}\big)$
$=\frac{\frac{\text{i}^2-1}{\text{i}}}{2}$
$=\frac{\text{i}^2-1}{2\text{i}}$
$=\frac{-1-1}{2\text{i}}$
$=\frac{-2}{-2\text{i}}$
$=\frac{-1}{\text{i}}$
$=\frac{-\text{i}}{\text{i}^2} \ \big(\because\text{i}^2=-1\big)$
$=\frac{-\text{i}}{-1}$
$=\text{i}$

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