<prompt> Acha Fzfracziz-i kwa nambari zote changamano znot i, na uruhusu z_nFz_n-1 kwa nambari zote chanya n. Kwa kuzingatia kwamba z_0frac 1137i, find z_2002.  <response>: Kurudia F mara chache, tunapata beginaligned FFz fracfracziz-iifraczi z-i-i fracziiz-izi-iz-i fraczizi1zi- zi-1 fracz1i1z-11-i fracz1i12z-1 cdot 2 fracz12iz-1 cdot 2 fracz1z-1i,FFFz fracfracz1z-1iifracz1z-1i-i frac fracz1z-11fracz1z-1-1 fracz1z-1z 1-z-1 z. mwishozilizopangiliwaKwa hivyo, z_k3 z_k kwa k. zote Tangu 2002 equiv 1 pmod3, basi tuna z_2002 z_1 fracz_0iz_0-i frac1137 2i1137 1274i. 