Question

Express following in $a+ib$ form
$\frac{1}{i}+\frac{2}{i^2}+\frac{3}{i^3}+\frac{5}{i^4}$

Answer 1

$\frac{1}{i}+\frac{2}{i^2}+\frac{3}{i^3}+\frac{5}{i^4}$

$=\frac{1}{i}+\frac{2}{-1}+\frac{3}{-i}+\frac{5}{1}$ [Since $i^2=-1,i^3=-i,i^4=1$]

$=-\frac{2}{i}+3$

$=3+2i$, which is the required $a+ib$ form.