等比数列an的前n项和为sn(sn≠0),则sn,s2n-sn,s3n-s2n成等比数列,公比为?等比数列an的前n项和为Sn(Sn≠0),则Sn,S2n-Sn,S3n-S2n成等比数列,公比为?

Question

等比数列an的前n项和为sn(sn≠0),则sn,s2n-sn,s3n-s2n成等比数列,公比为?
等比数列an的前n项和为Sn(Sn≠0),则Sn,S2n-Sn,S3n-S2n成等比数列,公比为?

Answer 1

Sn=a1+a2+a3+.+an S2n-Sn=a(n+1)+a(n+2)+a(n+3)+.+a2n =a1*q^n+a2*q^n+a3*q^n+.+an*q^n =(q^n)*(a1+a2+a3+.+an) =Sn*q^nS3n-S2n =a(2n+1)+a(2n+2)+a(2n+3)+.+a3n =a1 *q^2n+a2*q^2n+a3*q^2n+.+an*q^2n =(q^2n)（a1+a2+a3+.+an） =Sn*q^2nSn*(S3n-S2n)=(Sn^2)*(q^2n)(S2n-Sn）²=（ Sn^2)*(q^2n) Sn*S3n-S2n=（S2n-Sn）²,所以Sn,S2n-Sn,S3n-S2n成等比数列,公比为q^n.