Consider a sequence {a(n)} with a(1) = 2 and a(n) = (a(n-1))²/a(n-2) for all n ≥ 3.

It is known that a(2) and a(5) are positive integers and a(5) ≤ 2012.

What are the possible values of a(5)?

1.The recursion formula implies even integers only.

2. The 2012 limit causes the candidates  for a(2) not to exceed 10.

3. Therefore there are 5 solutions i.e. (a(2),a(5))= ((2,2),(4,32),(6,162),(8,512),(10,1250)).

4. Interesting fact : if a(2)=2   than all members are equal to 2, since
each of the members is 2^2/2.

 

Posted by Ady TZIDON on 2012-03-14 16:55:52