Today we share our analysis of the following problem, which kindly comes to us from Colombian engineer Ariel Nunez, a former Iberoamericano Math Olympiad Bronze medalist!
Given a real arithmetic progression , some terms are deleted to produce a geometric sequence with ratio . Find all possible values of .
Define for and . The geometric subsequence is for , where is the indexing sequence. Evaluating ,
Let . From the above recurrence, for some constant . From the initial condition ,
, which implies . Equating these expressions for ,
Since , each imposes a divisibility relationship between and .
From the case of , , so for some . Consequently,
and must both be positive for to be non-negative for all . By induction since , it follows that
This constraint cannot be satisfied if is irrational. Suppose , such that for non-commensurate . From the case of , implies that for some not divisible by . However, from the case of , can be rewritten as , which implies that is divisible by — which is a contradiction. Therefore, can only be a natural number. is achievable when , and all other natural numbers for are achievable when .
Do you have other approaches for solving the problem? Let us know!
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