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I've been practicing induction and I came across this problem:

Consider the following series, 1, 2, 3, 4, 5, 10,20, 40, ..., which starts as an arithmetic series, but after the first 5 terms becomes a geometric series. Prove that any positive integer can be written as a sum of distinct numbers from this series.

I've proved a similar problem to this concerning adding 4 and 5 cent stamps to come up with the correct postage. However, that problem doesn't stipulate that I use distinct numbers, so it is straightforward to prove with the well-ordering principle.

Can anyone help me figure out a way to handle this problem?

David J.
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    What do you know about the binary number system? That might seem like a strange question, but your problem features binary numbers in a fundamental, albeit obscured, way. – Arthur Sep 13 '19 at 13:35
  • @Arthur Now that you mention it, I know that any decimal number can be written in binary form, and that the 10, 20, 40... series can therefore be used to express any powers of 10 greater than 1. So I know that the statement is true. But does this constitute a proof? Wouldn't I have to prove this property of binary numbers? – David J. Sep 13 '19 at 13:42
  • That's why I asked. If that is something you can assume to be already known, then you can exploit that. If not, then yes, it would have to be proven. And if we're proving it from scratch anyway, there is no reason to go via binary. It will only muddle things. – Arthur Sep 13 '19 at 13:45

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