已知数列an是等比数列,前n项和为Sn,若S3,S9,S6成等差数列,求a2,a8,a5成等差数列

题目

已知数列an是等比数列,前n项和为Sn,若S3,S9,S6成等差数列,求a2,a8,a5成等差数列

答案

2S9=S3+S6
2a1(1-q^9)/(1-q)=a1(1-q^3)/(1-q)+a1(1-q^6)/(1-q)
2(1-q^9)=(1-q^3)+(1-q^6)
2(1-q^3)(1+q^3+q^6)=(1-q^3)+(1-q^3)(1+q^3)
若1-q^3=0,则q=1, 是常数列,则a2=a8=a5,等差
若1-q^3≠0
2+2q^3+2q^6=2+q^3
q^3+2q^6=0
q^3=-1/2
所以a5=a2*q^3=-a2/2
a8=a2*q^6=a2/4
则a2+a5=a2/2=2a8
所以a2,a8,a5成等差数列