The value of
Question:

The value of $\frac{i^{4 n+1}-i^{4 n-1}}{2}$ is _________________

Solution:

Let $z=\frac{i^{4 n+1}-i^{4 n-1}}{2}$

$=\frac{i^{4 n} \cdot i-i^{4 n}-i^{-1}}{2}$

$=\frac{i-i^{-1}}{2}$

$=\frac{i-\frac{1}{i}}{2}$

$=\frac{i^{2}-1}{2 i}$

$=-\frac{1-1}{2 i}$

$=-\frac{2}{2 i}$

$=-\frac{1}{i} \times \frac{i}{i}$

$=-\frac{i}{-1}$

i. e $\frac{i^{4 n+1}-i^{4 n-1}}{2}=i$

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