If n is any positive integer, write the value of

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

$=\frac{i-\frac{1}{i}}{2} \quad\left(\because i^{4 n}=1, i^{-1}=\frac{1}{i}\right)$

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

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

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

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

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

$=\frac{-i}{i^{2}} \quad\left(\because i^{2}=-1\right)$

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

$=i$