Question

已知数列{an}是等比数列,Sn是其前n项的和,a1,a7,a4成等差数列,求证2S3,S6,S12-S6 成等比数列?

Answer 1

a1,a7,a4成等差数列

2a7=a1+a4

2a1*q^6=a1(1+q^3)

2q^6-q^3-1=0

(2q^3+1)(q^3-1)=0

q^3=1,q=1

或:q^3=-1/2

若q=1,则2S3=S6=S12-S6=6a1,成等比数列

若:q^3=-1/2

2s3=2a1(q^3-1)/(q-1)

s6=a1(q^6-1)/(q-1)

s12-s6=a1(q^12-q^6)/(q-1)=a1q^6(q^6-1)/(q-1)

s6/2s3=(q^6-1)/2(q^3-1)=(q^3+1)/2=1/4

(s12-s6)/s6=q^6=(q^3)^2=1/4

∴s6/2s3=(s12-s6)/s6

∴ 2S3,S6,S12-S6 成等比数列