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.
ScanSolve AI Answer
Huu hapa uchambuzi wa swali la pili: 2. Nambari za 1^st, 5^th na 7^th za AP huunda nambari tatu za kwanza mfululizo za GP. Tofauti ya kawaida ya AP ni d na uwiano wa kawaida wa GP ni 32. a) Tafuta thamani za d na r. Step 1: Eleza nambari za AP. Acha nambari ya kwanza ya AP iwe a. Tofauti ya kawaida ya AP ni d. Nambari ya n^th ya AP inatolewa na T_n = a + (n-1)d. Hivyo, nambari za AP zinazohusika ni: T_1 = a T_5 = a + (5-1)d = a + 4d T_7 = a + (7-1)d = a + 6d Step 2: Tumia masharti ya GP. Nambari hizi T_1, T_5, T_7 huunda GP na uwiano wa kawaida r=32. Kwa GP, uwiano wa nambari mfululizo ni sawa na uwiano wa kawaida. (T_5)/(T_1) = r (a+4d)/(a) = 32 (Equation 1) (T_7)/(T_5) = r (a+6d)/(a+4d) = 32 (Equation 2) Step 3: Tatua Equation 1 kwa a kwa kutumia d. Kutoka Equation 1: a + 4d = 32a 4d = 32a - a 4d = 31a a = (4d)/(31) Step 4: Tatua Equation 2 na ubadilishe thamani ya a. Kutoka Equation 2: a + 6d = 32(a + 4d) a + 6d = 32a + 128d Panga upya ili kupata a na d: 6d - 128d = 32a - a -122d = 31a Sasa badilisha a = (4d)/(31) katika equation hii: -122d = 31((4d)/(31)) -122d = 4d -122d - 4d = 0 -126d = 0 d = 0 Step 5: Tafuta thamani ya a. Tumia a = (4d)/(31) na d=0: a = (4(0))/(31) a = 0 Kwa hivyo, d=0 na r=32 (imetolewa). a) Majibu: d = 0 r = 32 b) Tafuta jumla ya nambari 15 za kwanza za AP. Step 6: Tumia formula ya jumla ya AP. Jumla ya nambari n za kwanza za AP inatolewa na formula: S_n = (n)/(2)[2a + (n-1)d] Tunahitaji jumla ya nambari 15 za kwanza, hivyo n=15. Kutoka sehemu (a), tuna a=0 na d=0. Step 7: Badilisha maadili na ukokotoe jumla. S_15 = (15)/(2)[2(0) + (15-1)(0)] S_15 = (15)/(2)[0 + (14)(0)] S_15 = (15)/(2)[0 + 0] S_15 = (15)/(2)[0] S_15 = 0 b) Jibu: S_15 = 0 3 done, 2 left today. You're making progress.