关于等差数列的题设数列{An}的前n项和为Sn,已知A1=1,且满足3Sn^2=An(3Sn-1)(n≥2)(1)求证:

来源：学生作业帮编辑：百度作业网作业帮分类：数学作业时间：2024/06/24 08:29:45

关于等差数列的题
设数列{An}的前n项和为Sn,已知A1=1,且满足3Sn^2=An(3Sn-1)(n≥2)
(1)求证:数列{1/Sn}是等差数列
(2)设bn=Sn/3n+1,求数列{bn）的前n项和n

$关于等差数列的题设数列{An}的前n项和为Sn,已知A1=1,且满足3Sn^2=An(3Sn-1)(n≥2)(1)求证:$

(1)
3(Sn)^2=An(3Sn-1),
n≥2,An=Sn-S(n-1),
所以
3(Sn)^2=[Sn-S(n-1)](3Sn-1)
=3(Sn)^2-Sn-3SnS(n-1)+S(n-1)
3SnS(n-1)+Sn-S(n-1)=0,
SnS(n-1)≠0,两边同除以SnS(n-1)
1/Sn-1/S(n-1)=3,
1/S1=1/a1=1,
所以{1/Sn}是以1为首项,公差为3的等差数列.
(2)
由(1)知
1/Sn=1+3(n-1)=3n-2,
所以Sn=1/(3n-2),
Bn=Sn/(3n+1)
=1/[(3n+1)(3n-2)]
=1/3[1/(3n-2)-1/(3n+1)]
所以
Tn=B1+B2+...+Bn
=1/3[1-1/4+1/4-1/7+...-1/(3n-2)+1/(3n-2)-1/(3n+1)]
=1/3[1-1/(3n+1)]
=n/(3n+1)

关于等差数列的题设数列{An}的前n项和为Sn,已知A1=1,且满足3Sn^2=An(3Sn-1)(n≥2)(1)求证: 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}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 设数列an的前n项和为Sn,a1=1,an=(Sn/n)+2(n-1)(n∈N*) 求证:数列an为等差数列, 已知数列An的前n项和为Sn.且满足an+2Sn*Sn-1=0=2>,a1=1/2,求证1/Sn是等差数列,求通项an的已知数列an的前n项和为Sn,且满足Sn=Sn-1/2Sn-1+1,a1=1/2(1)求证:1/Sn是等差数列(2 已知数列{an}的前N项和为Sn 且an+1=Sn-n+3,a1=2,设Bn=n/Sn-n+2前N项和为Tn 求证Tn 设数列{an}的前n项和为Sn,满足2Sn=an+1-2^n+1+1,且a1,a2+5.a3成等差数列设数列[an}的前n项和为Sn,已知a1=1且满足3Sn的平方=an(3Sn-1),1求{1/Sn}为等差数列 1.求{ 已知数列an的前n项和为Sn,且an+2Sn*Sn-1=0,a1=1/2,求证1/SN是等差数列,求数列SN的的通项公式已知数列{an}的前n项和为Sn,且满足an+2Sn+Sn-1=0(n≥2),a1+1/2