If I start giving you some numbers like:

or

That’s a **sequence.** You can take a sequence and make the **sequence of averages.** The average of is . The average of and is . The average of , and is . and so on. So the sequence of averages for

is

,

and the sequence of averages for

is

.

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 that answer to:

5) Can you come up with a sequence of positive rationals that itself hits every 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.

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