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The known sequence {an} satisfies A1, a2-a1, a3-a2 an-an-1,… Is the prime minister is 1, the common ratio is one third of the equal ratio sequence 1 The known sequence {an} satisfies A1, a2-a1, a3-a2 an-an-1,… The prime minister is 1, the common ratio is one-third of the equal ratio series 1. Find the general term formula of sequence {an} 2. If BN = (2n-1) an, find the first n terms and Sn of sequence {BN}
(1)∵a1,a2-a1,a3-a2,… an-an-1,… The prime minister is 1, the common ratio is one-third of the equal ratio series
∴an－an-1＝(1/3)^(n－1)
an-1－an-2＝(1/3)^(n－2)
… …
a2－a1＝1/3
∴an－a1＝1/3＋(1/3)²＋… ＋(1/3)^(n－1)＝[1－(1/3)^(n－1)]/2
∴an＝[3－(1/3)^(n－1)]/2
(2)bn＝(2n－1)an＝3/2(2n－1)－(2n－1)(1/3)^(n－1)/2
Sn＝3/2×[1＋3＋… ＋(2n－1)]－1/2×[1＋3×1/3＋5×(1/3)²＋… ＋(2n－1)(1/3)^(n－1)]
Let t = 1 + 3 × 1 / 3 + 5 × (1 / 3) & 178; + ＋(2n－1)(1/3)^(n－1)
∴T/3＝ 1/3＋3×(1/3)²＋… ＋(2n－3)(1/3)^(n－1)＋(2n－1)(1/3)^n
∴2T/3＝1＋2[1/3＋(1/3)²＋… ＋(1/3)^(n－1)]－(2n－1)(1/3)^n
＝1＋[1－(1/3)^(n－1)]－(2n－1)(1/3)^n＝2－(1/3)^(n－1)－(2n－1)(1/3)^n
∴Sn＝3/2×[1＋(2n－1)]n/2＋3/2×[2－(1/3)^(n－1)－(2n－1)(1/3)^n]
＝3/2×[n²＋3－2(n＋1)(1/3)^n]
RELATED INFORMATIONS
- 1. The known sequence {an} satisfies A1, A2-A3, a4-a3 , an-a (n-1) is the expression of finding an from the equal ratio sequence (1) with Prime Minister of 1 and common ratio of 1 / 3 (2) If BN = (2n-1) an, find the first n terms and Sn of the sequence {BN} It's wrong. (1) it should be A1, a2-a1, a3-a2 ，
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- 11. If the sequence {an} satisfies that A1, a2-a1, a3-a2,... Are equal ratio sequences with 1 as prime minister and 3 as common ratio, then an______
- 12. Given that {an} is an equal ratio sequence, and a1 + a3 = 10, A4 + A6 = 5 / 4, find A4 and S5
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- 14. Given the equal ratio sequence a1 + A6 = 11, A3 * A4 = 32 / 9, find an
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- 16. In the equal ratio sequence, Sn = a1 + A2 + a3 + +An, given A3 = 2s2 + 1, A4 = 2s3 + 1, what is the common ratio?
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