等比数列{an}的前n项和为Sn,若S3+3S2=0,则公比q=

Question

如题所述

Answer 1

1
S3=a1(1-q^3)/(1-q)，S2=a1(1-q^2)/(1-q)
故:a1(1-q^3)/(1-q)=-3*a1(1-q^2)/(1-q)
即:1+q+q^2=-3-3q，即:q^2+4q+4=(q+2)^2=0，故:q=-2
2
|2a-b|^2=(2a-b)·(2a-b)=4|a|^2+|b|^2-4a·b=4+|b|^2-4a·b
=4+|b|^2-4|b|cos(π/4)=4+|b|^2-2sqrt(2)|b|=10
故:|b|^2-2sqrt(2)|b|-6=0，故:|b|=(2sqrt(2)±4sqrt(2))/2
即:|b|=3sqrt(2)或-sqrt(2)(不和题意，舍去)
即:|b|=3sqrt(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