b) The 3rd, 5th and 8th terms of another AP form the first three consecutive terms. If the common difference…

Question

b) The 3rd, 5th and 8th terms of another AP form the first three consecutive terms. If the common difference of AP is 3, find: i) The 1st term or GP.

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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: A₃ = A + (3−1)D = A + 2D = A + 2(3) = A + 6; Neno la 5: A₅ = A + (5−1)D = A + 4D = A + 4(3) = A + 12; Neno la 8: A₈ = A + (8−1)D = A + 7D = A + 7(3) = A + 21 Maneno haya huunda GP, kwa hivyo: G₁ = A + 6; G₂ = A + 12; G₃ = A + 21 Step 1: Tumia sifa ya GP kwamba uwiano wa kawaida (r) kati ya maneno mfululizo ni sawa. r = (G₂)/(G₁) = (G₃)/(G₂) 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); A² + 24A + 144 = A² + 21A + 6A + 126; A² + 24A + 144 = A² + 27A + 126 Toa A² 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 (G₁). Neno la kwanza la GP ni G₁ = A₃ = A + 6. G₁ = 6 + 6; G₁ = 12 Neno la kwanza la GP ni 12. b) ii) Tunahitaji kupata jumla ya maneno 9 ya kwanza ya GP (S₉). Kutoka sehemu (i), tunajua neno la kwanza la GP ni G₁ = 12. Step 1: Pata uwiano wa kawaida (r) wa GP. r = (G₂)/(G₁) = (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 = (G₁(rⁿ − 1))/(r − 1) (kwa r > 1). Badilisha maadili: G₁ = 12, r = 3/2, na n = 9. S₉ = (12((3/2)⁹ − 1))/(3/2 − 1) Step 3: Rahisisha mlinganyo. S₉ = (12((3⁹)/(2⁹) − 1))/(1/2); S₉ = 24(19683/512 − 1); S₉ = 24((19683 − 512)/(512)); S₉ = 24(19171/512) Step 4: Fanya hesabu na toa jibu kwa idadi muhimu ya tarakimu (significant figures). S₉ = (24 × 19171)/(512); S₉ = (3 × 19171)/(64); S₉ = 57513/64 S₉ ≈ 898.640625 Kuzungusha kwa tarakimu 3 muhimu (3 significant figures): S₉ ≈ 899 Jumla ya maneno 9 ya kwanza ya GP ni 899.

b) The 3rd, 5th and 8th terms of another AP form the first three consecutive terms. If the common difference of AP is 3, find: i) The 1st term or GP.

Step-by-step solution

Tumia sifa ya GP kwamba uwiano wa kawaida ( $r$ ) kati ya maneno mfululizo ni sawa.

Tatua mlinganyo kwa $A$ .

Pata neno la kwanza la GP ( $G_1$ ).

Pata uwiano wa kawaida ( $r$ ) wa GP.

Tumia fomula ya jumla ya maneno $n$ ya GP, $S_n = \frac{G_1(r^n - 1)}{r - 1}$ (kwa $r > 1$ ).

Rahisisha mlinganyo.

Fanya hesabu na toa jibu kwa idadi muhimu ya tarakimu (significant figures).

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b) The 3rd, 5th and 8th terms of another AP form the first three consecutive terms. If the common difference of AP is 3, find: i) The 1st term or GP.

Step-by-step solution

Tumia sifa ya GP kwamba uwiano wa kawaida (rrr) kati ya maneno mfululizo ni sawa.

Tatua mlinganyo kwa AAA.

Pata neno la kwanza la GP (G1G_1G1​).

Pata uwiano wa kawaida (rrr) wa GP.

Tumia fomula ya jumla ya maneno nnn ya GP, Sn=G1(rn−1)r−1S_n = \frac{G_1(r^n - 1)}{r - 1}Sn​=r−1G1​(rn−1)​ (kwa r>1r > 1r>1).

Rahisisha mlinganyo.

Fanya hesabu na toa jibu kwa idadi muhimu ya tarakimu (significant figures).

Still stuck on this one?

More Mathematics Questions

Tumia sifa ya GP kwamba uwiano wa kawaida ( $r$ ) kati ya maneno mfululizo ni sawa.

Tatua mlinganyo kwa $A$ .

Pata neno la kwanza la GP ( $G_1$ ).

Pata uwiano wa kawaida ( $r$ ) wa GP.

Tumia fomula ya jumla ya maneno $n$ ya GP, $S_n = \frac{G_1(r^n - 1)}{r - 1}$ (kwa $r > 1$ ).