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  4. Let z=a( cos (π/5)+i sin (π/5)), a ∈ R ,|a|<1, then S=z2015+z2016+z2017+ ldots equals

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Q. Let $z=a\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right), a \in R ,|a|<1$, then $S=z^{2015}+z^{2016}+z^{2017}+\ldots$ equals

Complex Numbers and Quadratic Equations

Solution:

Solution: We have $|z|=|a|<1$, thus
$S =\frac{z^{2015}}{1-z} $
$\text { But } \quad z^{2015} =a^{2015}[\cos (403 \pi)+i \sin (403 \pi)]=-a^{2015} $
$\therefore S =\frac{a^{2015}}{z-1}$