What is the last number of terms of a G.P. 2+4+8 ... that will give a sum greater than 1,500,000?

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Hatua ya 1: Tambua neno la kwanza (a) na uwiano wa kawaida (r) wa GP. Kutoka kwa mfuatano 2, 4, 8, : Neno la kwanza, a = 2. Uwiano wa kawaida, r = 4/2 = 2. Hatua ya 2: Andika fomula ya jumla ya maneno n ya GP. Kwa kuwa r > 1, fomula ya jumla ya maneno n (S_n) ni: S_n = (a(rⁿ − 1))/(r − 1) Hatua ya 3: Badilisha thamani za a na r kwenye fomula. S_n = (2(2ⁿ − 1))/(2 − 1); S_n = (2(2ⁿ − 1))/(1); S_n = 2(2ⁿ − 1) Hatua ya 4: Weka usawa kulingana na sharti kwamba jumla ni kubwa kuliko 1,500,000. 2(2ⁿ − 1) > 1,500,000 Hatua ya 5: Tatua usawa kwa n. Gawanya pande zote mbili kwa 2: (2(2ⁿ − 1))/(2) > (1,500,000)/(2) 2ⁿ − 1 > 750,000 Ongeza 1 kwa pande zote mbili: 2ⁿ > 750,000 + 1 2ⁿ > 750,001 Ili kutatua kwa n, chukua logariti ya pande zote mbili (kwa mfano, logariti ya msingi 10 au asili): log(2ⁿ) > log(750,001) n log(2) > log(750,001) n > (log(750,001))/(log(2)) Tumia kikokotoo kupata thamani za logariti: log(750,001) ≈ 5.87506 log(2) ≈ 0.30103 n > 5.87506/0.30103 n > 19.516 Hatua ya 6: Tafuta nambari kamili ndogo zaidi ya maneno n inayokidhi usawa. Kwa kuwa n lazima iwe nambari kamili (idadi ya maneno), na n lazima iwe kubwa kuliko 19.516, nambari kamili ndogo zaidi inayokidhi sharti hili ni 20. Nambari ndogo zaidi ya maneno ambayo itatoa jumla kubwa kuliko 1,500,000 ni 20. 20

What is the last number of terms of a G.P. 2+4+8 ... that will give a sum greater than 1,500,000?

Answer

Step-by-step solution

Tambua neno la kwanza ( $a$ ) na uwiano wa kawaida ( $r$ ) wa GP.

Andika fomula ya jumla ya maneno $n$ ya GP.

Badilisha thamani za $a$ na $r$ kwenye fomula.

Weka usawa kulingana na sharti kwamba jumla ni kubwa kuliko 1,500,000.

Tatua usawa kwa $n$ .

Tafuta nambari kamili ndogo zaidi ya maneno $n$ inayokidhi usawa.

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What is the last number of terms of a G.P. 2+4+8 ... that will give a sum greater than 1,500,000?

Answer

Step-by-step solution

Tambua neno la kwanza (aaa) na uwiano wa kawaida (rrr) wa GP.

Andika fomula ya jumla ya maneno nnn ya GP.

Badilisha thamani za aaa na rrr kwenye fomula.

Weka usawa kulingana na sharti kwamba jumla ni kubwa kuliko 1,500,000.

Tatua usawa kwa nnn.

Tafuta nambari kamili ndogo zaidi ya maneno nnn inayokidhi usawa.

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Tambua neno la kwanza ( $a$ ) na uwiano wa kawaida ( $r$ ) wa GP.

Andika fomula ya jumla ya maneno $n$ ya GP.

Badilisha thamani za $a$ na $r$ kwenye fomula.

Tatua usawa kwa $n$ .

Tafuta nambari kamili ndogo zaidi ya maneno $n$ inayokidhi usawa.