This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Hili ni swali kuhusu mfululizo wa kijiometri (geometric sequence). Tutatumia formula ya jumla ya vigezo vya mfululizo wa kijiometri. Formula ya jumla ya n vigezo vya kwanza vya mfululizo wa kijiometri ni: S_n = (a(r^n - 1))/(r - 1) ambapo a ni kigezo cha kwanza, r ni uwiano wa kawaida (common ratio), na n ni idadi ya vigezo. Step 1: Tumia taarifa uliyopewa kupata kigezo cha kwanza (a). Tunajua S_3 = 14 na r = -3. 14 = (a((-3)^3 - 1))/(-3 - 1) Step 2: Rahisisha na utatue kwa a. 14 = (a(-27 - 1))/(-4) 14 = (a(-28))/(-4) 14 = 7a a = (14)/(7) a = 2 Step 3: Sasa tumia thamani ya a uliyopata na r=-3 ili kupata jumla ya vigezo 5 vya kwanza (S_5). S_5 = (a(r^5 - 1))/(r - 1) S_5 = (2((-3)^5 - 1))/(-3 - 1) Step 4: Fanya hesabu. S_5 = (2(-243 - 1))/(-4) S_5 = (2(-244))/(-4) S_5 = (-488)/(-4) S_5 = 122 Jumla ya vigezo 5 vya kwanza ni: 122 3 done, 2 left today. You're making progress.