数列{an}的首项a1=1,前n项和Sn满足关系:3tSn-(2t+3)Sn-1=3t(t>0,n=2,3,4,5,).求和:b1b2-b2b3+b3b4-b4b5+...+(-1)^n+1bnbn+1.(1)数列{an}是等比数列(要有推理过程);(2)设数列{an}的公比为f(t),做数列{bn},
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数列{an}的首项a1=1,前n项和Sn满足关系:3tSn-(2t+3)Sn-1=3t(t>0,n=2,3,4,5,).求和:b1b2-b2b3+b3b4-b4b5+...+(-1)^n+1bnbn+1.(1)数列{an}是等比数列(要有推理过程);(2)设数列{an}的公比为f(t),做数列{bn},
数列{an}的首项a1=1,前n项和Sn满足关系:3tSn-(2t+3)Sn-1=3t(t>0,n=2,3,4,5,).
求和:b1b2-b2b3+b3b4-b4b5+...+(-1)^n+1bnbn+1.
(1)数列{an}是等比数列(要有推理过程);(2)设数列{an}的公比为f(t),做数列{bn},使b1=1,bn=f(1/b(n-1))(n=2,…),求数列{bn}
以上是(3)
数列{an}的首项a1=1,前n项和Sn满足关系:3tSn-(2t+3)Sn-1=3t(t>0,n=2,3,4,5,).求和:b1b2-b2b3+b3b4-b4b5+...+(-1)^n+1bnbn+1.(1)数列{an}是等比数列(要有推理过程);(2)设数列{an}的公比为f(t),做数列{bn},
(1)a1=1
3tSn- (2t+3) Sn-1=3t (t >0,n=2,3,4,5,) (1).
3tSn-1-(2t+3)Sn-2=3t (t >0,n=2,3,4,5,) (2).
(1)-(2)得:
3tSn- (2t+3) Sn-1= 3tSn-1-(2t+3)Sn-2
整理得:
3tSn-3tSn-1=(2t+3) Sn-1-(2t+3)Sn-2
即 3t(Sn-Sn-1)=(2t+3) (Sn-1-Sn-2)
3t an= ( 2t+3) an-1
因为t不等于0
所以 an/an-1=2t+3/3t
当n=1时也成立
所以,数列{an}是等比数列
(2)由(1)可知数列{an}的公比f(t)=2t+3/3t
bn=f(1/b(n-1))(n=2,3,4,…)
代入整理得:
bn = 3 b(n-1)+2
bn +1= 3 (b(n-1)+1)
{bn +1}是等比数列
即 bn = 3^n-1 -1 (n=2,3,4,…)
b1=1