已知数列{an}中,a1=1,a(n+1)=an/(2an+1) 已知数列{an}中，a1=1,a(n+1)=an/(...
\u5df2\u77e5\u6570\u5217{an}\u4e2d\uff0ca1=1,a(n+1)=an/(2an+1\uff09\uff0c\u6c42an\u7684\u901a\u9879\u516c\u5f0f\u3002a(n+1)=an/(2an +1)
1/a(n+1)=(2an +1)/an =1/an +2
1/a(n+1)-1/an=2\uff0c\u4e3a\u5b9a\u503c\u3002
1/a1=1/1=1
\u6570\u5217{1/an}\u662f\u4ee51\u4e3a\u9996\u9879\uff0c2\u4e3a\u516c\u5dee\u7684\u7b49\u5dee\u6570\u5217\u3002
1/an =1+2(n-1)=2n-1
an=1/(2n-1)
\u6570\u5217{an}\u7684\u901a\u9879\u516c\u5f0f\u4e3aan=1/(2n-1)\u3002
a(n+1)=an/(2an
+1)
1/a(n+1)=(2an
+1)/an
=1/an
+2
1/a(n+1)-1/an=2\u5b9a\u503c
1/a1=1/1=1
\u6570\u5217{1/an}1\u9996\u98792\u516c\u5dee\u7b49\u5dee\u6570\u5217
1/an
=1+2(n-1)=2n-1
an=1/(2n-1)
\u6570\u5217{an}\u901a\u9879\u516c\u5f0fan=1/(2n-1)
A(n+1)+1=2An+2
（A(n+1)+1）/（An+1）=2
所以{An+1}是等比数列
所以{2An+2}是等比数列
②{An+1}是首项为2公比为2的等比数列，An+1=2^n
所以An=2^n-1
③Sn=2(1-2^n)/(1-2)-n=2^(n+1)-2-n
1.
证：
a(n+1)=an/(2an+1)
1/a(n+1)=(2an+1)/an=1/an +2
1/a(n+1)-1/an=2，为定值。
1/a1=1/1=1，数列{1/an}是以1为首项，2为公差的等差数列。
2.
1/an=1+2(n-1)=2n-1
an=1/(2n-1)
bn=ana(n+1)=[1/(2n-1)][1/(2n+1)]=(1/2)[1/(2n-1)-1/(2n+1)]
Sn=b1+b2+...+bn
=(1/2)[1-1/3+1/3-1/5+...+1/(2n-1)-1/(2n+1)]
=(1/2)[1-1/(2n+1)]
=n/(2n+1)
Sn>1005/2012
n/(2n+1)>1005/2012
2n>1005
n>502.5，又n为正整数，n≥503，n的最小值是503。
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