The following problem is from Martin Liebeck’s book, A Concise Introduction to Pure Mathematics (2nd ed., Chapman and Hall, Boca Raton 2006). The book is based on an introductory course Liebeck teaches to students of Mathematics-based degrees at Imperial College. The exercise below uses no fancy advanced results, just clear thinking. It is, as Liebeck says, for fun.
Let be a sequence of positive integers such that for all ,
Find . What can you say about the sequence in general?
Hint: start by finding , then , etc.