设等比数列{an}的公比q≠1,若{an+c}也是等比数列,则c=_.

问题描述:

设等比数列{an}的公比q≠1,若{an+c}也是等比数列,则c=______.

∵{an+c}是等比数列
∴(a1+c)(a3+c)=(a2+c)2
即a1a3+c(a1+a3)+c2=a22+2a2c+c2
∵a1a3=a22
∴(a1+a3)c=2a2c
即a1c(1+q2)=2a1qc
移项ca1(1-2q+q2)=0
∴a1≠0,1-2q+q2≠0,则c=0
故答案为0.