Problem: If a1,a2,a3,… is a sequence of positive numbers such that an+2= anan+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,a1a2,a1a22,a12a23. If these terms are in geometric progression, then the ratios of successive terms must be equal: a1=a2=a1a2. 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, ….