STD 11 - 4.1 complex nubers — Maths STD 11 Science — Question

Now the given sum

$S = \sum\limits_{k = 1}^6 {\left( {\sin \frac{{2\pi k}}{7} - i\cos \frac{{2\pi k}}{7}} \right)} $
=$\sum\limits_{k = 1}^6 {\left[ {( - i)\left( {\cos \frac{{2\pi k}}{7} + i\sin \frac{{2\pi k}}{7}} \right)} \right]} $

$ = ( - i)\sum\limits_{k = 1}^6 {\left( {\cos \frac{{2\pi k}}{7} + i\sin \frac{{2\pi k}}{7}} \right) = ( - i)\sum\limits_{k = 1}^6 {{z^k}} } $

Which is a G.P. of which the first term is $z$, number of terms is $6$ and the common ratio is $z = \cos \frac{{2\pi }}{7} + i\sin \frac{{2\pi }}{7} \ne 1$ So summing up the G.P.,

we have $S = ( - i)\frac{{z(1 - {z^6})}}{{1 - z}} = ( - i)\frac{{z - {z^7}}}{{1 - z}} = ( - i)\frac{{z - 1}}{{1 - z}} = i$