已知数列{an}的前n项和为Sn,且满足a1=1,Sn-Sn-1=2SnSn-1(n≥2). (1)数列{1/Sn}是否为等差数列?请证明你的结论; (2)求数列{an}的前n项和Sn.
问题描述:
已知数列{an}的前n项和为Sn,且满足a1=1,Sn-Sn-1=2SnSn-1(n≥2).
(1)数列{
}是否为等差数列?请证明你的结论;1 Sn
(2)求数列{an}的前n项和Sn.
答
(1)∵Sn-Sn-1=2SnSn-1∴1Sn−1−1Sn=2即1Sn−1Sn−1=−2(常数)∴{1Sn}为等差数列 (2)∵1Sn=1S1+(n−1)(−2)=1−2n+2=−2n+3∴Sn=1−2n+3....