设数列{an}满足a1=0,2an+1=1+anan+1 (1)求证:为等差数列,并求通项公式an (2)记 ,求数列{bn}的前n
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设数列{an}满足a1=0,2an+1=1+anan+1(1)求证:为等差数列,并求通项公式an(2)记,求数列{bn}的前n设数列{an}满足a1=0,2an+1=1+anan+1(1)求证:为等差
设数列{an}满足a1=0,2an+1=1+anan+1 (1)求证:为等差数列,并求通项公式an (2)记 ,求数列{bn}的前n
设数列{an}满足a1=0,2an+1=1+anan+1 (1)求证:为等差数列,并求通项公式an (2)记 ,求数列{bn}的前n
设数列{an}满足a1=0,2an+1=1+anan+1 (1)求证:为等差数列,并求通项公式an (2)记 ,求数列{bn}的前n
2a(n+1)=1+a(n)a(n+1)
移向整理得到2a(n+1)-a(n)a(n+1)=1,即[2-a(n)]a(n+1)=1
都除以2-a(n)得到a(n+1)=1/[2-a(n)]
设c(n)=a(n)/[a(n)-1]
所以c(n+1)=a(n+1)/[a(n+1)-1]=1/[2-a(n)]/{[1/[2-a(n)]-1}=1/[a(n)-1]
所以c(n+1)-cn=1/[a(n)-1]-a(n)/[a(n)-1]=-1
所以数列{a(n)/a(n)-1]}为公差-1的等差数列.
因为c1=a(1)/[a(1)-1]=0,所以cn=c1+(n-1)(-1)=1-n
即a(n)/[a(n)-1]=1-n,
化简得到an=(n-1)/n
2an+1=1+anan+1
anan-2an+1=0
(an-1)^2=0
an=1
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