公比不为-1的等比数列的前n项和为Sn,则Sn,S2n-Sn,S3n-S2n仍成等比数列,则其公比为多少?

Question

公比不为-1的等比数列的前n项和为Sn,则Sn,S2n-Sn,S3n-S2n仍成等比数列,则其公比为多少?

Answer 1

Sn=a1+a2+a3+.+an=A
S2n-Sn=a(n+1)+a(n+2)+a(n+3)+.+a(2n)
=a1*q^n+a2*q^n+a3*q^n+.+a(n)*q^n
=A*q^n
S3n-S2n=a(2n+1)+a(2n+2)+a(2n+3)+.+a(3n)
=a1*q^(2n)+a2*q^(2n)+a3*q^(2n)+.+a(n)*q^(2n)
=A*q^(2n)
所以
（S3n-S2n）/(S2n-Sn)=(S2n-Sn)/Sn
Sn,S2n-Sn,S3n-S2n成等比数列,其公比是q^n