## Puzzle: Liebeck’s sequence

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)