a) The first term of an Arithmetic progression (AP) is 3. The sum of the first 6 terms of the AP is 78. i ) Find the common difference of AP. ii ) Given that the sum of the first n terms of the AP is 406, find n. b ) The 2nd,4th and 7 th terms of another AP form the first three terms of a Geometric Progression (GP).If the common difference of the AP is 2, find; i )The first term of the GP; ii )The sum of the first 7 terms of the GP, to 4 significant figures.

Question

Asked on August 30, 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 · 2026-08-30T09:00:01.287Z

a) i) Ili kupata tarakimu ya kawaida (common difference) ya AP: Step 1: Tumia formula ya jumla ya n vigezo vya AP, Sₙ = (n/2)[2a + (n-1)d]. Tunapewa a = 3, n = 6, na S₆ = 78. S₆ = (6/2)[2(3) + (6-1)d] 78 = 3[6 + 5d] Step 2: Gawanya pande zote mbili kwa 3. 78/3 = 6 + 5d 26 = 6 + 5d Step 3: Toa 6 kutoka pande zote mbili na kisha gawanya kwa 5. 26 - 6 = 5d 20 = 5d d = 20/5 d = 4 Tarakimu ya kawaida (common difference) ni 4. ii) Ili kupata n wakati Sₙ = 406: Step 1: Tumia formula ya jumla ya n vigezo vya AP tena, Sₙ = (n/2)[2a + (n-1)d]. Tunajua a = 3 na d = 4. 406 = (n/2)[2(3) + (n-1)4] 406 = (n/2)[6 + 4n - 4] 406 = (n/2)[4n + 2] Step 2: Panua na kurahisisha mlinganyo. 406 = 2n² + n 2n² + n - 406 = 0 Step 3: Tatua mlinganyo wa quadratic kwa kutumia formula ya quadratic au kwa kuweka vigezo. Kwa kutumia kuweka vigezo: 2n² + 29n - 28n - 406 = 0 n(2n + 29) - 14(2n + 29) = 0 (n - 14)(2n + 29) = 0 Step 4: Pata thamani za n. n - 14 = 0 ⟹ n = 14 2n + 29 = 0 ⟹ n = -29/2 Kwa kuwa n lazima iwe nambari chanya kamili (idadi ya vigezo), tunachagua n = 14. n = 14 b) i) Ili kupata kigezo cha kwanza (first term) cha GP: Step 1: Acha kigezo cha kwanza cha AP kiwe A. Tarakimu ya kawaida ya AP ni d_AP = 2. Vigezo vya AP ni: T₂ = A + (2-1)d_AP = A + 2 T₄ = A + (4-1)d_AP = A + 3(2) = A + 6 T₇ = A + (7-1)d_AP = A + 6(2) = A + 12 Hivi ndivyo vigezo vitatu vya kwanza vya GP: g₁ = A + 2 g₂ = A + 6 g₃ = A + 12 Step 2: Katika GP, uwiano wa kawaida (common ratio) ni sawa. Hivyo, g₂/g₁ = g₃/g₂. (A + 6)/(A + 2) = (A + 12)/(A + 6) Step 3: Panua na tatua kwa A. (A + 6)(A + 6) = (A + 12)(A + 2) A² + 12A + 36 = A² + 2A + 12A + 24 A² + 12A + 36 = A² + 14A + 24 Step 4: Rahisisha mlinganyo. 12A + 36 = 14A + 24 36 - 24 = 14A - 12A 12 = 2A A = 12/2 A = 6 Step 5: Kigezo cha kwanza cha GP ni g₁ = A + 2. g₁ = 6 + 2 g₁ = 8 Kigezo cha kwanza cha GP ni 8. ii) Ili kupata jumla ya vigezo 7 vya kwanza vya GP: Step 1: Kwanza, pata uwiano wa kawaida (common ratio) r wa GP. r = g₂/g₁ = (A + 6)/(A + 2) = (6 + 6)/(6 + 2) = 12/8 = 3/2 r = 1.5 Step 2: Tumia formula ya jumla ya n vigezo vya GP, Sₙ = a(rⁿ - 1)/(r - 1). Tunajua a = 8 (kigezo cha kwanza cha GP), r = 1.5, na n = 7. S₇ = 8((1.5)⁷ - 1)/(1.5 - 1) S₇ = 8((1.5)⁷ - 1)/0.5 Step 3: Kokotoa (1.5)⁷. (1.5)⁷ ≈ 17.0859375 Step 4: Endelea kukokotoa S₇. S₇ = 8(17.0859375 - 1)/0.5 S₇ = 8(16.0859375)/0.5 S₇ = 128.6875/0.5 S₇ = 257.375 Step 5: Zungusha jibu kwa tarakimu 4 muhimu (4 significant figures). S₇ = 257.4 Jumla ya vigezo 7 vya kwanza vya GP ni 257.4. Got more? Send 'em 📸

Accepted Answer · 2026-08-30T09:00:01.287Z

a) i) Ili kupata tarakimu ya kawaida (common difference) ya AP: Step 1: Tumia formula ya jumla ya n vigezo vya AP, Sₙ = (n/2)[2a + (n-1)d]. Tunapewa a = 3, n = 6, na S₆ = 78. S₆ = (6/2)[2(3) + (6-1)d] 78 = 3[6 + 5d] Step 2: Gawanya pande zote mbili kwa 3. 78/3 = 6 + 5d 26 = 6 + 5d Step 3: Toa 6 kutoka pande zote mbili na kisha gawanya kwa 5. 26 - 6 = 5d 20 = 5d d = 20/5 d = 4 Tarakimu ya kawaida (common difference) ni 4. ii) Ili kupata n wakati Sₙ = 406: Step 1: Tumia formula ya jumla ya n vigezo vya AP tena, Sₙ = (n/2)[2a + (n-1)d]. Tunajua a = 3 na d = 4. 406 = (n/2)[2(3) + (n-1)4] 406 = (n/2)[6 + 4n - 4] 406 = (n/2)[4n + 2] Step 2: Panua na kurahisisha mlinganyo. 406 = 2n² + n 2n² + n - 406 = 0 Step 3: Tatua mlinganyo wa quadratic kwa kutumia formula ya quadratic au kwa kuweka vigezo. Kwa kutumia kuweka vigezo: 2n² + 29n - 28n - 406 = 0 n(2n + 29) - 14(2n + 29) = 0 (n - 14)(2n + 29) = 0 Step 4: Pata thamani za n. n - 14 = 0 ⟹ n = 14 2n + 29 = 0 ⟹ n = -29/2 Kwa kuwa n lazima iwe nambari chanya kamili (idadi ya vigezo), tunachagua n = 14. n = 14 b) i) Ili kupata kigezo cha kwanza (first term) cha GP: Step 1: Acha kigezo cha kwanza cha AP kiwe A. Tarakimu ya kawaida ya AP ni d_AP = 2. Vigezo vya AP ni: T₂ = A + (2-1)d_AP = A + 2 T₄ = A + (4-1)d_AP = A + 3(2) = A + 6 T₇ = A + (7-1)d_AP = A + 6(2) = A + 12 Hivi ndivyo vigezo vitatu vya kwanza vya GP: g₁ = A + 2 g₂ = A + 6 g₃ = A + 12 Step 2: Katika GP, uwiano wa kawaida (common ratio) ni sawa. Hivyo, g₂/g₁ = g₃/g₂. (A + 6)/(A + 2) = (A + 12)/(A + 6) Step 3: Panua na tatua kwa A. (A + 6)(A + 6) = (A + 12)(A + 2) A² + 12A + 36 = A² + 2A + 12A + 24 A² + 12A + 36 = A² + 14A + 24 Step 4: Rahisisha mlinganyo. 12A + 36 = 14A + 24 36 - 24 = 14A - 12A 12 = 2A A = 12/2 A = 6 Step 5: Kigezo cha kwanza cha GP ni g₁ = A + 2. g₁ = 6 + 2 g₁ = 8 Kigezo cha kwanza cha GP ni 8. ii) Ili kupata jumla ya vigezo 7 vya kwanza vya GP: Step 1: Kwanza, pata uwiano wa kawaida (common ratio) r wa GP. r = g₂/g₁ = (A + 6)/(A + 2) = (6 + 6)/(6 + 2) = 12/8 = 3/2 r = 1.5 Step 2: Tumia formula ya jumla ya n vigezo vya GP, Sₙ = a(rⁿ - 1)/(r - 1). Tunajua a = 8 (kigezo cha kwanza cha GP), r = 1.5, na n = 7. S₇ = 8((1.5)⁷ - 1)/(1.5 - 1) S₇ = 8((1.5)⁷ - 1)/0.5 Step 3: Kokotoa (1.5)⁷. (1.5)⁷ ≈ 17.0859375 Step 4: Endelea kukokotoa S₇. S₇ = 8(17.0859375 - 1)/0.5 S₇ = 8(16.0859375)/0.5 S₇ = 128.6875/0.5 S₇ = 257.375 Step 5: Zungusha jibu kwa tarakimu 4 muhimu (4 significant figures). S₇ = 257.4 Jumla ya vigezo 7 vya kwanza vya GP ni 257.4. Got more? Send 'em 📸

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