已知数列{An}的前n项和为Sn,且满足An+2SnSn-1=0(n大于等于2),A1=1/2.(1)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+2SnSn-1=0(n大于等于2),A1=1/2.(1)求证:{1/Sn}是等差数列,(2)求An的

Question

已知数列{An}的前n项和为Sn,且满足An+2SnSn-1 = 0(n大于等于2),A1 = 1/2 .(1)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+2SnSn-1 = 0(n大于等于2),A1 = 1/2 .
(1)求证:{ 1/Sn }是等差数列,
(2)求An的表达式.
(3)求证：S1的平方+S2的平方+S3的平方+……+Sn的平方≤1/2-1/4n

Answer 1

(1) An=Sn-S(n-1)
Sn-S(n-1)+2SnS(n-1)=0
1/S(n-1)-1/S(n)+2=0
1/S(n)=1/S(n-1)+2
{ 1/Sn }是等差数列,d=2.1/S1=1/A1=2,1/Sn=2n
(2)Sn=1/(2n)
An=Sn-S(n-1)=1/(2n)-1/(2(n-1))=-1/[2n(n-1)]
(3)S1^2+...Sn^2=1/4[1+1/2^2+...1/n^2]