Question

等比数列{an}的前n项和为Sn,已知2S1,S3,S2成等差数列

(1)求数列{an}的公比q
(2)若a1-a3=3,求Sn

Answer 1

1.
2S3=2S1+S2
2a1(1+q+q^2)=2a1+a1(1+q)
2(1+q+q^2)=2+1+q
2q^2+q-1=0
q=-1或q=1/2

2.
a1-a3=3
a1-a1q^2=3
a1(1-q^2)=3
当q=-1时，a1不存在；
当q=1/2时，
a1=3/(1-q^2)=4
sn=a1(1-q^n)/(1-q)
=4[1-(1/2)^n]/(1-1/2)
=8-(1/2)^(n-3)