Question

在数列{an}中a1=1,当n≥2时,an,Sn,Sn-1/2成等比数列.求数列｛an｝的表达式

Answer 1

an,Sn,Sn-1/2成等比数列

an(Sn-1/2)=Sn^2

a2(S2-1/2)=S2^2

a2(a2+1/2)=(a2+1)^2

a2=-2/3

a3(S3-1/2)=S3^2

a3(a3-1/6)=(a3+1/3)^2

a3=-2/33

[Sn-S(n-1)](Sn-1/2)=Sn^2

-(1/2)Sn-S(n-1)Sn+(1/2)S(n-1)=0

-(1/2)Sn+(1/2)S(n-1)=S(n-1)Sn

1/Sn-1/S(n-1)=-2

1/Sn=(1/S2)+(-2)(n-2)

=[1/(1-2/3)]+(-2)(n-2)

=3+(-2)(n-2)

=-2n+7

Sn=1/(-2n+7)

S(n-1)=1/(-2n+5)

an=Sn-S(n-1)=1/(-2n+7)-1/(-2n+5)

an=1/(-2n+7)-1/(-2n+5);