已知数列{an}满足a1=3,anan-1=2an-1-1,(1)求证{1/an-1}是等差数列

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已知数列{an}满足a1=3,anan-1=2an-1-1,(1)求证{1/an-1}是等差数列已知数列{an}满足a1=3,anan-1=2an-1-1,(1)求证{1/an-1}是等差数列已知数列

已知数列{an}满足a1=3,anan-1=2an-1-1,(1)求证{1/an-1}是等差数列
已知数列{an}满足a1=3,anan-1=2an-1-1,(1)求证{1/an-1}是等差数列

已知数列{an}满足a1=3,anan-1=2an-1-1,(1)求证{1/an-1}是等差数列
(1)由题意知a(n-1)≠0,∴an=2-1/a(n-1)=2a(n-1)-1/a(n-1) ∴ a(n-1)-1=a(n-1)-1/a(n-1)
∴1/a(n-1)=a(n-1)/a(n-1)-1=a(n-1)-1+1/a(n-1)-1=1+1/a(n-1)-1 ∴1/an-1_1/a(n-1)-1=1
又 1/a1-1=1/2
∵1为常数 ∴1/an-1是以1/2为首项,1位公差的等差数列
(2)(这里就不全部打出来了,毕竟比较麻烦)先根据一算出an=2n+1/2n-1,an再带进去得到bn=1/(2n+1)(2n-1),再代入Sn用裂项相消法,即bn=1/2(1/2n-1_1/2n+1),算出Sn=n/2n+1

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