Question

等比数列【an】的前n项和为sn 若s3+3s2=0 则公比q=

Answer 1

Sn=[a1*（1-q^n）/(1-q)]
S3=a1*(1-q^3)/(1-q)
S2=a1*(1-q^2)/(1-q)
S3+3S2=0
a1*(1-q^3)/(1-q)+3*a1*(1-q^2)/(1-q)=0
因为是等比数列，所以a1不为0 q不为1 即1-q不为0
所以1-q^3+3(1-q^2)=0
(q+2)^2*(q-1)=0
q不为1
所以q只能等于-2

Answer 2

q=1时(an≠0)
S3=3a1
S2=2a1
S3+3S2=0
a1=0(舍)
q=-1时
S3=a1
S2=0
S3+2S2=0
a1=0(舍)
|q|≠1时且q≠0
S3=a1(1-q^3)/(1-q)=a1(1+q+q²)
S2=a1(1-q²)/(1-q)=a1(1+q)
S3+3S2=0
a1(4+4q+q²)=0
4+4q+q²=0

q=-2