若数列{An}满足A1=1,A(n+1)=An/(2An + 1)

问题描述:

若数列{An}满足A1=1,A(n+1)=An/(2An + 1)
(n∈N+)
(1)求A2,A3
(2)判断数列{1/An}是否成等差数列,并说明理由.
(3)求证:1/3≤A1A2+A2A3+A3A4+.+AnA(n+1)

1) 1/3,1/52)倒数变换一下即可证明从该步骤得到an=1/(2n-1)3) T=(1/1*1/3+1/3*1/5+1/5*1/7+……+[1/(2n-3)][1/(2n-1)]=1/2(1-1/3+1/3-1/5+1/5-1/7+……+1/(2n-5)-1/(2n-3)+1/(2n-3)-1/(2n-1)=1/2(1-1/(2n-1) =n/(2n...