## Sequences of Rationals (rerun)

Here’s a rerun of an earlier post.  It ends with an open question that I hope someone can help me out with.

If I start giving you some numbers like:

$1, 2, 3, 4, 5, \ldots$

or

$1, \frac{1}{2},\frac{1}{4},\ldots$

That’s a sequence. You can take a sequence and make the sequence of averages. The average of $1$ is $1$. The average of $1$ and $2$ is $\frac{3}{2}$. The average of $1$, $2$ and $3$ is $2$. and so on. So the sequence of averages for

$1, 2, 3, 4, 5, \ldots$

is

$1, \frac{3}{2}, 2, \frac{5}{2}, 3\ldots$,

and the sequence of averages for

$1, \frac{1}{2},\frac{1}{4},\ldots$

is

$1, \frac{3}{4}, \frac{7}{12},\ldots$.

Now for some questions (the answers are known)

1) Can you come up with a sequence of rational numbers such that the sequence of averages hits every rational number?

2) Can you come up with a sequence of positive rationals such that the sequence of averages hits every positive rational number?

3) Can you come up with a sequence of positive rationals such that the sequence of averages hits every positive rational exactly once?

4) How about a sequence of unique positive rationals (no repeats) whose sequence of averages hits every positive rational exactly once?

Finally, one that I don’t know the answer to:

5) Can you come up with a sequence of positive rationals that itself hits every positive rational exactly once and whose sequence of averages also hits every positive rational exactly once?

Please let me know if you get number (5). Also let me know if you are stuck on any of them.