<prompt> Thamani ya jumla ni nini sum_z frac1left1 - zright2 , ,ambapo z inatofautiana juu ya masuluhisho yote 7 halisi na yasiyo halisi ya mlingano z7 -1?  <response>: Kwa kuwa z7 -1, z7 1. Kisha z7 1, hivyo z 1. Kisha z overlinez z2 1, hivyo overlinez frac1z. Kwa hiyo,anzaalignfrac11 - z2 frac11 - zoverline1 - z frac11 - z1 - overlinez frac11 - z1 - frac1z fracz1 - zz - 1 -fraczz - 12.endalignHebu z frac1w 1. Kisha-fraczz - 12 -fracfrac1w 1frac1w2 -w - w 2.Kutoka z7 -1,kushoto frac1w 1 kulia7 -1.Kisha 1 w7 -w7. Kupanua, tunapata2w7 7w6 21w5 35w4 35w3 21w2 7w 1 0. Acha mizizi ya z7 -1 iwe z_1, z_2, dots, z_7, na acha w_k iwe thamani inayolingana ya z_k, i.e. z_k frac1w_k 1. Kishasum_k 17 frac11 - z_k2 sum_k 17 -w_k - w_k2.Kwa fomula za Vieta, w_1 w_2 dots w_7 -frac72 na w_1 w_2 w_1 w_3 dots w_6 w_7 frac212. Kuweka sawa mlinganyo w_1 w_2 dots w_7 -frac72, tunapataw_12 w_22 vitone w_72 2w_1 w_2 w_1 w_3 vitone w_6 w_7 frac494.Kishaw_12 w_22 vitone w_72 frac494 - 2w_1 w_2 w_1 w_3 vitone w_6 w_7 frac494 - 2 cdot frac212 -frac354.Kwa hiyo,sum_k 17 -w_k - w_k2 frac72 frac354 frac494. 