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  4. The value of (1/i)+(1/i2)+(1/i3)+⋅s+(1/i102) is

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Q. The value of $\frac{1}{i}+\frac{1}{i^{2}}+\frac{1}{i^{3}}+\cdots+\frac{1}{i^{102}}$ is

KEAMKEAM 2013Complex Numbers and Quadratic Equations

Solution:

$\frac{1}{i}+\frac{1}{i^{2}}+\frac{1}{i^{3}}+\ldots+\frac{1}{i^{102}}$
$\therefore \, S_{n}=\frac{\frac{1}{i}\left\{1-\left(\frac{1}{i}\right)^{102}\right\}}{1-\left(\frac{1}{i}\right)}=\frac{(1+1)}{i-1}$
$\begin{pmatrix}\because i^{4}=1 \\ \text { and } i^{2}=-1\end{pmatrix}$
$=\frac{2(i+1)}{i^{2}-1}=-(i+1)=-1-i$