若{an}为公比不为1的等比数列,
设公比为q,则Sn=a1(q^n-1)/(q-1)=:a(q^n-1)
若数列{an}的前n项和为Sn=a(b^n-1)
则当n>=2时,an=Sn-Sn-1=a(b^n-1)-a(b^(n-1)-1)=a(b-1)b^(n-1)
当n=1时,a1=S1=a(b-1)
则an=a(b-1)b^(n-1),n>=1
即{an}为等比数列
从而,{an}为等比数列当且仅当Sn=a(b^n-1)