Inspired by a recent post over at Foxmaths!, here’s an interesting challenge question for you to think about:

Suppose I give you the sequence of integers , and so on, where denotes the greatest integer less than or equal to *x*—in other words, it means to round down. So the first number in the sequence would be , the next number would be , and so on. Given this sequence, what can you learn about (assuming that you didn’t already know anything about it)?

A more general question: given the sequence of integers for *k* = 1,2,3…, what can you learn about *r*?

The answer has many interesting connections to the theory of irrational numbers and continued fractions.

Well, the obvious thing one can learn given just |(10^n)r| is the first n digits of the decimal expansion of r. I presume there’s supposed to be more to learn?

Well done! But you can actually learn the decimal expansion of r faster than that.

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