Just want to record a solution to a relatively simple Putnam problem here. The problem is the following:
Show that a sequence satisfies the condition . Prove that diverges.
I tried attacking it with the usual methods of showing that a sequence diverges, but nothing seemed to work. However, every good Olympiad worth its salt asks for some experimentation, which will then yield important observations/patterns. I did the same here.
Without loss of generality, let . Then . Hence, . Similary, and . Therefore, . Continuing this pattern, we observe that for all , . Hence , which can be thought of as , diverges.