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b) i) Tunapewa kwamba neno la 3, la 5, na la 8 la AP nyingine huunda maneno matatu ya kwanza mfululizo ya GP. Tofauti ya kawaida ya AP hii ni D = 3. Acha neno la kwanza la AP liwe A. Maneno ya AP ni: Neno la 3: A_3 = A + (3-1)D = A + 2D = A + 2(3) = A + 6 Neno la 5: A_5 = A + (5-1)D = A + 4D = A + 4(3) = A + 12 Neno la 8: A_8 = A + (8-1)D = A + 7D = A + 7(3) = A + 21 Maneno haya huunda GP, kwa hivyo: G_1 = A + 6 G_2 = A + 12 G_3 = A + 21 Step 1: Tumia sifa ya GP kwamba uwiano wa kawaida (r) kati ya maneno mfululizo ni sawa. r = (G_2)/(G_1) = (G_3)/(G_2) Badilisha maneno ya GP: (A + 12)/(A + 6) = (A + 21)/(A + 12) Step 2: Tatua mlinganyo kwa A. Panua pande zote mbili: (A + 12)^2 = (A + 6)(A + 21) A^2 + 24A + 144 = A^2 + 21A + 6A + 126 A^2 + 24A + 144 = A^2 + 27A + 126 Toa A^2 kutoka pande zote mbili na panga upya: 144 - 126 = 27A - 24A 18 = 3A A = (18)/(3) A = 6 Step 3: Pata neno la kwanza la GP (G_1). Neno la kwanza la GP ni G_1 = A_3 = A + 6. G_1 = 6 + 6 G_1 = 12 Neno la kwanza la GP ni 12. b) ii) Tunahitaji kupata jumla ya maneno 9 ya kwanza ya GP (S_9). Kutoka sehemu (i), tunajua neno la kwanza la GP ni G_1 = 12. Step 1: Pata uwiano wa kawaida (r) wa GP. r = (G_2)/(G_1) = (A + 12)/(A + 6) Badilisha A = 6: r = (6 + 12)/(6 + 6) = (18)/(12) r = (3)/(2) Step 2: Tumia fomula ya jumla ya maneno n ya GP, S_n = (G_1(r^n - 1))/(r - 1) (kwa r > 1). Badilisha maadili: G_1 = 12, r = (3)/(2), na n = 9. S_9 = (12((3)/(2))^9 - 1)(3)/(2) - 1 Step 3: Rahisisha mlinganyo. S_9 = (12(3^9)/(2^9) - 1)(1)/(2) S_9 = 24((19683)/(512) - 1) S_9 = 24((19683 - 512)/(512)) S_9 = 24((19171)/(512)) Step 4: Fanya hesabu na toa jibu kwa idadi muhimu ya tarakimu (significant figures). S_9 = (24 × 19171)/(512) S_9 = (3 × 19171)/(64) S_9 = (57513)/(64) S_9 ≈ 898.640625 Kuzungusha kwa tarakimu 3 muhimu (3 significant figures): S_9 ≈ 899 Jumla ya maneno 9 ya kwanza ya GP ni 899. That's 2 down. 3 left today — send the next one.