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 ,

and

Find . What can you say about the sequence in general?

*Hint: start by finding , then , etc.*

(MG)

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