数列{an}前n项和为Sn,且满足a1=1,an=-SnS(n-1),(n≥2),求S5
问题描述：
数列{an}前n项和为Sn,且满足a1=1,an=-SnS(n-1),(n≥2),求S5
答
S1=a1=1,a2=-S2*S1,a2=-1/2,S2=1/2,a3=-S3*S2,a3=-(a3+S2)*S2,a3=-1/6
S3=1/3,a4=-(a4+S3)*S3,a4=-1/12,S4=1/4,a5=-(a5+S4)*S4,a5=-1/20,
S5=1/5
答
an=-SnS(n-1)
Sn-S(n-1)=-SnS(n-1) (两边同除以SnS(n-1) )
1/S(n-1)-1/Sn=-1
1/Sn-1/S(n-1)=1
所以{1/Sn}是以1/S1=1/a1=1为首相d=1公差的等差数列
1/Sn=1+(n-1)=n
所以Sn=1/n
S5=1/5