数列an中an大于0且数列an乘以an+1是公比为q（q大于0）的等比数列,满足(anan+1)+(an+1an+2)大于an+2an+3,求公比q的取值范围.

数列an中an大于0且数列an乘以an+1是公比为q（q大于0）的等比数列,满足(anan+1)+(an+1an+2)大于an+2an+3,求公比q的取值范围.
数列an中an大于0且数列an乘以an+1是公比为q（q大于0）的等比数列,满足(anan+1)+(an+1an+2)大于an+2an+3,
求公比q的取值范围.

数列an中an大于0且数列an乘以an+1是公比为q（q大于0）的等比数列,满足(anan+1)+(an+1an+2)大于an+2an+3,求公比q的取值范围.
令Bn=AnAn+1,则数列{Bn}是公比为q（q大于0）的等比数列
An+1An+2=Bn+1=Bn*q
An+2An+3=Bn+2=Bn*q²
Bn+Bn+1＞Bn+2
Bn+Bn*q＞Bn*q²
因为Bn＞0
所以1+q＞q²且q＞0
所以0＜q＜(1+√5)/2

0＜q＜(1+√5)/2