依题意可知Ai=ai•a(i+1),
则:A(i+1)=a(i+1)•a(i+2),
必要性:
由于{An}为等比数列
则
A(i+1)/Ai=a(i+2)/ai
=q(q为常数)
由于:i=1,2,3...,即i为任意正整数
则a1,a3,…,a(2n-1),…和a2,a4,…,a(2n),…均是等比数列,且公比均为q;
充分性:
a1,a3,…a(2n-1),…和a2,a4,…a(2n),…均是等比数列,且公比相同时
可得:a(i+2)/ai=t (t为公比)
则
A(i+1)/Ai
=(a(i+1)*a(i+2))/(ai*a(i+1))
=a(i+2)/ai
=t
为常数,
即{An}为等比数列
故{An}为等比数列的充要条件是a1,a3,…,a2n-1,…和a2,a4,…,a2n,…均是等比数列,且公比相同.
故选D