Question

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

Answer 1

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