Sn²=an(Sn-1/2)
Sn²=[Sn-Sn-1](Sn-1/2)
整理,得
Sn+2SnSn-1-Sn-1=0
等式两边同除以SnSn-1
1/Sn-1+2-1/Sn=0
1/Sn-1/Sn-1=2,为定值.
1/S1=1/a1=1/1=1
数列{1/Sn}是以1为首项,2为公差的等差数列.
1/Sn=1+2(n-1)=2n-1
Sn=1/(2n-1)
an=Sn-Sn-1=1/(2n-1)-1/(2n-3)
n=1时,a1=2,不满足.
数列{an}的通项公式为
an=1 n=1
1/(2n-1)-1/(2n-3) n≥2