0
$\begingroup$

If we choose randomly two integers $(a, b)$ from a sequence of $k$ distinct positive integers then it is always possible to find a third element in the sequence such that there is an arithmetic progression between $a$, $b$ and $c$.

If such a sequence exists what is the maximum value of $k$?

  • 1
    What sort of "*sequence of size $k$*" are we talking about? I could say take the zero sequence of length $k$ for any arbitrary $k$, $(0,0,0,\dots,0)$ taking any two you can have an arithmetic progression $0,0,0$ – 2017-02-14
  • 0
    a sequence of $k$ distincts positive integers – 2017-02-14
  • 0
    So, let me try to understand your question a bit better... you are asking - Find the largest value of $k$ for which there exists a sequence of length $k$ of distinct positive integers with the property that for any two elements selected from the sequence there exists a third element in the sequence such that the three elements form an arithmetic progression. It doesn't bother you that there exist other sequences such that there exist no arithmetic progressions within it, just that there exists at least one sequence with the aforementioned property, – 2017-02-14
  • 0
    Yes this is exactly what i am asking for. – 2017-02-14

0 Answers 0