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.


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One Response to Puzzle: Liebeck’s sequence

  1. Pingback: Liebeck’s sequence: a solution | Degree of Freedom

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