Question

The value of (i⁵ + i⁶ + i⁷ + i⁸ + i⁹) / (1 + i) is
(a) $\frac{1}{2}(1+i)$
(b) $\frac{1}{2}(1-i)$
(c) 1
(d) $\frac{1}{2}$

Answer 1

(a) $\frac{1}{2}(1+i)$

$$\begin{gathered} \frac{i^5+i^6+i^7+i^8+i^9}{1+i} \\ =\frac{i-1-i+1+i}{1+i}\left[\mathrm{As},i^5=i,i^6=-1,i^7=-i,i^8=1,i^9=i\right] \\ =\frac{i}{i+1} \\ =\frac{i}{i+1}\times \frac{i-1}{i-1} \\ =\frac{i\left(i-1\right)}{i^2-1} \\ =\frac{i^2-i}{-2} \\ =\frac{1}{2}\left(1+i\right) \end{gathered}$$