<prompt> Mlolongo wa a_1,b_1, a_2,b_2, a_3,b_3, ldets wa pointi katika ndege ya kuratibu hutoshelezaa_n 1, b_n 1 sqrt 3a_n - b_n, sqrt 3b_n a_nkwa n 1,2,3, dots. Tuseme kuwa a_100,b_100 2,4. a_1 b_1 ni nini? Eleza jibu lako kwa kutumia nukuu maalum.  <response>: Hebu z_n a_n b_n i. Kishaanzaalignz_n 1 a_n sqrt3 - b_n b_n sqrt3 a_n i a_n sqrt3 i b_n i sqrt3 - 1 a_n sqrt3 i b_n i sqrt3 i sqrt3 ia_n b_n i sqrt3 i z_n.endalignKwa hivyo, z_100 sqrt3 i99 z_1. Ili kutathmini usemi huu, tunaandikasqrt3 i 2 cdot left fracsqrt32 frac12 i kulia 2 jina la operetacis 30 circ.Kishasqrt3 i99 299 jina la operetacis 2970circ 299 jina la operetacis 90circ 299 i.Kwa kuwa z_100 2 4i,z_1 frac2 4i299 i frac4 - 2i299 frac2 - i298, hivyoa_1 b_1 frac2298 - frac1298 frac1298. 