已知数列{An}的前n项和Sn满足Sn=1-2/3An.

来源：学生作业帮编辑：大师作文网作业帮分类：数学作业时间：2024/11/19 09:15:48

已知数列{An}的前n项和Sn满足Sn=1-2/3An.
求lim(A1S1+A2S2+…AnSn).

n=1时 S1=1-(2/3)A1 解得A1=3/5
n>1时 S(n-1)=1-(2/3)A(n-1)
An=Sn-S(n-1)=-(2/3)An+(2/3)A(n-1)
所以An=(2/5)*A(n-1)
所以{An}是公比为2/5的等比数列
故An=(3/5)*(2/5)^(n-1)
所以Sn=1-(2/3)An
=1-(2/5)*(2/5)^(n-1)
=1-(2/5)^n
AnSn=An*[1-(2/3)An]
=An-(2/3)An²
=An-(2/3)*(3/5)²*(2/5)^(2n-2)
=An-(6/25)*(4/25)^(n-1)
所以A1S1+A2S2+...+AnSn
=(A1+A2+.+An)-(6/25)*[1+(4/25)+...+(4/25)^(n-1)]
=Sn-(6/25)*[1-(4/25)^n]/(1-4/25)
=1-(2/5)^n-2/7+(2/7)*(4/25)^n
=5/7-(2/5)^n-(4/25)^n
故lim(A1S1+A2S2+…AnSn) (n→∞)
=5/7

已知数列{an}a1=2前n项和为Sn 且满足Sn Sn-1=3an 求数列{an}的通项公式an 数列an的前n项和Sn满足：Sn=2an-3n 已知数列{an}的前n项和为Sn,通项an满足Sn+an=1/2(n2+3n-2),求通项公式an. 已知数列{an}的前n项的和Sn,满足6Sn=an2+3an+2且an>0.(1)求首项a1;(2)证明{an}是已知数列{a}的前n项和Sn,通项an满足Sn+an=1/2(n^2+3n-2),求通项公式an 已知数列an首相a1=3,通项an和前n项和SN之间满足2an=Sn*Sn-1（n大于等于2）已知数列{an}的前n项和为Sn,且满足Sn=2an-1,n为正整数,求数列{an}的通项公式an 已知数列{an}的前n项和为Sn,且满足Sn=2an-1(n属于正整数),求数列{an}的通项公式an 已知数列An的前n项和Sn满足An+2Sn*Sn-1=0,n大于等于2,A1=1/2,求An. 已知数列{an}的前n项和为Sn,且满足an+2Sn*Sn-1=0,a1=1/2.求证：{1/Sn}是等差数列已知数列{an}的前n项和为Sn,且满足Sn=Sn-1/2Sn-1 +1,a1=2,求证{1/Sn}是等差数列已知数列{an}的前n项和sn满足:s1=1,3Sn=(n+2)an.求an的通项