Problem: If a1​,a2​,a3​,… is a sequence of positive numbers such that an+2​= an​an+1​ for all positive integers n, then the sequence a1​,a2​,a3​,… is a geometric progression
Answer Choices:
A. for all positive values of a1​ and a2​
B. if and only if a1​=a2​
C. if and only if a1​=1
D. if and only if a2​=1
E. if and only if a1​=a2​=1
Solution:
The second through the fifth terms of {an​} are a2​,a1​a2​,a1​a2​2,a1​2a2​3. If these terms are in geometric progression, then the ratios of successive terms must be equal: a1​=a2​=a1​a2​. Since a1​ and a2​ are positive, it is necessary that a1​=a2​=1. Conversely, if a1​=a2​=1, then {an​} is the geometric progression 1, 1, 1, ….