a1=2,an+an-1=(n/(an-an-1))+2,求1/(a1-1)^2+1/(a2-1)^2+...+1/(a

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/11/09 00:38:19

a1=2,an+an-1=(n/(an-an-1))+2,求1/(a1-1)^2+1/(a2-1)^2+...+1/(an-1)^2的极限
Thanks very much~

an+an-1=(n/(an-an-1))+2 去分母整理得（an-1）^2=(a(n-1)-1)^2+n
=>
（an-1）^2=(a(n-1)-1)^2+n
=(a(n-2)-1)^2+n-1 +n
...
= 1+2+ ...+n = n(n+1)/2
1/（an-1）^2 = 1/(n(n+1)/2) = 2/n - 2/(n+1)
==>
1/(a1-1)^2+1/(a2-1)^2+...+1/(an-1)^2 = 2 - 2/(n+1) --> 2

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