a) The first term of an Arithmetic progression (AP) is 3. The sum of the first 6 terms of the AP is 78. i )…

Question

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.

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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.

Step-by-step solution

Part a(i)· 3 steps

Tumia formula ya jumla ya n vigezo vya AP, Sₙ = (n/2)[2a + (n-1)d].

Gawanya pande zote mbili kwa 3.

Toa 6 kutoka pande zote mbili na kisha gawanya kwa 5.

Part a(ii)· 4 steps

Tumia formula ya jumla ya n vigezo vya AP tena, Sₙ = (n/2)[2a + (n-1)d].

Panua na kurahisisha mlinganyo.

Tatua mlinganyo wa quadratic kwa kutumia formula ya quadratic au kwa kuweka vigezo.

Pata thamani za n.

Part b(i)· 5 steps

Acha kigezo cha kwanza cha AP kiwe A.

Katika GP, uwiano wa kawaida (common ratio) ni sawa.

Panua na tatua kwa A.

Rahisisha mlinganyo.

Kigezo cha kwanza cha GP ni g₁ = A + 2.

Part b(ii)· 5 steps

Kwanza, pata uwiano wa kawaida (common ratio) r wa GP.

Tumia formula ya jumla ya n vigezo vya GP, Sₙ = a(rⁿ - 1)/(r - 1).

Kokotoa (1.5)⁷.

Endelea kukokotoa S₇.

Zungusha jibu kwa tarakimu 4 muhimu (4 significant figures).

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