The 2nd, 4th and 7th terms of an AP are the first 3 consecutive terms of a GP. If the common difference of the AP is 2, find: the common ratio of the GP. the sum of the first eight terms of the GP.

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Asked on September 6, 2026Mathematics

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.

Accepted Answer · 2026-09-06T18:32:55.561Z

Huu hapa ni utatuzi wa swali la 15: Kwanza, tutaandika maneno ya AP na GP. Acha A iwe neno la kwanza la AP na D iwe tofauti ya kawaida. Tunapewa D = 2. Maneno ya AP ni: a_n = A + (n-1)D Maneno ya 2^nd, 4^th na 7^th ya AP ni: a_2 = A + (2-1)D = A + D = A + 2 a_4 = A + (4-1)D = A + 3D = A + 3(2) = A + 6 a_7 = A + (7-1)D = A + 6D = A + 6(2) = A + 12 Maneno haya matatu (a_2, a_4, a_7) ndiyo maneno matatu ya kwanza mfululizo ya GP. Kwa hivyo, acha g_1 = A + 2, g_2 = A + 6, na g_3 = A + 12. Kwa GP, uwiano wa maneno mfululizo ni sawa: (g_2)/(g_1) = (g_3)/(g_2) Hii inamaanisha (g_2)^2 = g_1 · g_3. Step 1: Tafuta thamani ya A. Tutabadilisha maneno ya GP kwenye fomula ya uwiano: (A + 6)^2 = (A + 2)(A + 12) Panua pande zote mbili: A^2 + 12A + 36 = A^2 + 12A + 2A + 24 A^2 + 12A + 36 = A^2 + 14A + 24 Toa A^2 kutoka pande zote mbili: 12A + 36 = 14A + 24 Panga upya ili kutatua A: 36 - 24 = 14A - 12A 12 = 2A A = 6 Kwa hivyo, neno la kwanza la AP ni 6. Step 2: Tafuta maneno ya kwanza ya GP. Sasa tunaweza kupata maneno ya kwanza ya GP: g_1 = A + 2 = 6 + 2 = 8 g_2 = A + 6 = 6 + 6 = 12 g_3 = A + 12 = 6 + 12 = 18 a) Tafuta uwiano wa kawaida wa GP. Step 3: Kokotoa uwiano wa kawaida R. Uwiano wa kawaida R ni (g_2)/(g_1): R = (12)/(8) = (3)/(2) Unaweza pia kuthibitisha kwa (g_3)/(g_2): R = (18)/(12) = (3)/(2) Uwiano wa kawaida wa GP ni (3)/(2). b) Tafuta jumla ya maneno nane ya kwanza ya GP. Step 4: Kokotoa jumla ya maneno nane ya kwanza ya GP. Fomula ya jumla ya maneno n ya kwanza ya GP ni S_n = (G(R^n - 1))/(R - 1), ambapo G ni neno la kwanza na R ni uwiano wa kawaida. Hapa, G = 8, R = (3)/(2), na n = 8. S_8 = (8 ( (3)/(2))^8 - 1 )(3)/(2) - 1 Kwanza, kokotoa ((3)/(2))^8: ((3)/(2))^8 = (3^8)/(2^8) = (6561)/(256) Kisha, kokotoa kigawanyo: (3)/(2) - 1 = (3)/(2) - (2)/(2) = (1)/(2) Sasa badilisha hizi kwenye fomula ya jumla: S_8 = (8 ( 6561)/(256) - 1 )(1)/(2) S_8 = (8 ( 6561 - 256)/(256) )(1)/(2) S_8 = (8 ( 6305)/(256) )(1)/(2) Kurahisisha: S_8 = 8 × (6305)/(256) × 2 S_8 = (16 × 6305)/(256) S_8 = (6305)/(16) Jumla ya maneno nane ya kwanza ya GP ni (6305)/(16) au 394.0625. Tuma swali linalofuata 📸

Accepted Answer · 2026-09-06T18:32:55.561Z

Huu hapa ni utatuzi wa swali la 15: Kwanza, tutaandika maneno ya AP na GP. Acha A iwe neno la kwanza la AP na D iwe tofauti ya kawaida. Tunapewa D = 2. Maneno ya AP ni: a_n = A + (n-1)D Maneno ya 2^nd, 4^th na 7^th ya AP ni: a_2 = A + (2-1)D = A + D = A + 2 a_4 = A + (4-1)D = A + 3D = A + 3(2) = A + 6 a_7 = A + (7-1)D = A + 6D = A + 6(2) = A + 12 Maneno haya matatu (a_2, a_4, a_7) ndiyo maneno matatu ya kwanza mfululizo ya GP. Kwa hivyo, acha g_1 = A + 2, g_2 = A + 6, na g_3 = A + 12. Kwa GP, uwiano wa maneno mfululizo ni sawa: (g_2)/(g_1) = (g_3)/(g_2) Hii inamaanisha (g_2)^2 = g_1 · g_3. Step 1: Tafuta thamani ya A. Tutabadilisha maneno ya GP kwenye fomula ya uwiano: (A + 6)^2 = (A + 2)(A + 12) Panua pande zote mbili: A^2 + 12A + 36 = A^2 + 12A + 2A + 24 A^2 + 12A + 36 = A^2 + 14A + 24 Toa A^2 kutoka pande zote mbili: 12A + 36 = 14A + 24 Panga upya ili kutatua A: 36 - 24 = 14A - 12A 12 = 2A A = 6 Kwa hivyo, neno la kwanza la AP ni 6. Step 2: Tafuta maneno ya kwanza ya GP. Sasa tunaweza kupata maneno ya kwanza ya GP: g_1 = A + 2 = 6 + 2 = 8 g_2 = A + 6 = 6 + 6 = 12 g_3 = A + 12 = 6 + 12 = 18 a) Tafuta uwiano wa kawaida wa GP. Step 3: Kokotoa uwiano wa kawaida R. Uwiano wa kawaida R ni (g_2)/(g_1): R = (12)/(8) = (3)/(2) Unaweza pia kuthibitisha kwa (g_3)/(g_2): R = (18)/(12) = (3)/(2) Uwiano wa kawaida wa GP ni (3)/(2). b) Tafuta jumla ya maneno nane ya kwanza ya GP. Step 4: Kokotoa jumla ya maneno nane ya kwanza ya GP. Fomula ya jumla ya maneno n ya kwanza ya GP ni S_n = (G(R^n - 1))/(R - 1), ambapo G ni neno la kwanza na R ni uwiano wa kawaida. Hapa, G = 8, R = (3)/(2), na n = 8. S_8 = (8 ( (3)/(2))^8 - 1 )(3)/(2) - 1 Kwanza, kokotoa ((3)/(2))^8: ((3)/(2))^8 = (3^8)/(2^8) = (6561)/(256) Kisha, kokotoa kigawanyo: (3)/(2) - 1 = (3)/(2) - (2)/(2) = (1)/(2) Sasa badilisha hizi kwenye fomula ya jumla: S_8 = (8 ( 6561)/(256) - 1 )(1)/(2) S_8 = (8 ( 6561 - 256)/(256) )(1)/(2) S_8 = (8 ( 6305)/(256) )(1)/(2) Kurahisisha: S_8 = 8 × (6305)/(256) × 2 S_8 = (16 × 6305)/(256) S_8 = (6305)/(16) Jumla ya maneno nane ya kwanza ya GP ni (6305)/(16) au 394.0625. Tuma swali linalofuata 📸

The 2nd, 4th and 7th terms of an AP are the first 3 consecutive terms of a GP. If the common difference of the AP is 2, find: the common ratio of the GP. the sum of the first eight terms of the GP.

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The 2nd, 4th and 7th terms of an AP are the first 3 consecutive terms of a GP. If the common difference of the AP is 2, find: the common ratio of the GP. the sum of the first eight terms of the GP.

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