Question

已知数列{an}是等比数列,公比q,Sn是其前n项和,证明Sn,S2n-Sn,S3n-S2n成等比数列

Answer 1

令1+q+q²+.+q^(n-1)=A

Sn=a1（1+q+q²+.+q^(n-1)）=a1*A

S2n-Sn=a1(q^n+q^(n+1)+.+q^(2n-1))=a1q^(n)（1+q+q²+.+q^(n-1)）=a1q^(n)*A

S3n-S2n=a1(q^2n+q^(2n+1)+...+q^(3n-1))=a1q^(2n)（1+q+q²+.+q^(n-1)）=a1q^(2n)*A

[S2n-Sn]²=a1*a1q^(2n)*A*A=(a1*A)*[a1q^(2n)*A]=Sn*[S3n-S2n]

Sn,S2n-Sn,S3n-S2n成等比数列