Puzzle: Liebeck’s sequence

Posted on September 30, 2011 by strathmaths

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 $a_1, \, a_2, \ldots$ be a sequence of positive integers such that for all $n \geq 1$ ,

$a_{n+1} > a_n$ and $a_{a_n} = 3n.$

Find $a_{100}$ . What can you say about the sequence in general?

Hint: start by finding $a_1$ , then $a_2$ , etc.

(MG)

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