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Aryabhatta
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 100th number in sequence   « on: Jun 16th, 2007, 9:40am » Quote Modify

Consider the sequence 1, 3, 4, 9, 10 ...

which is such that each number is either a power of 3 or the sum of distinct powers of 3. The numbers are arranged in increasing order.

Without using a computer/calculator, find the 100th number of this sequence.
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Grimbal
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 Re: 100th number in sequence   « Reply #1 on: Jun 16th, 2007, 9:56am » Quote Modify

Can I write the number in base 3, for added difficulty?
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Aryabhatta
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 Re: 100th number in sequence   « Reply #2 on: Jun 16th, 2007, 10:19am » Quote Modify

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thecuriousone
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 Re: 100th number in sequence   « Reply #3 on: Jun 16th, 2007, 12:02pm » Quote Modify

 hidden: is it 981?     This is what I observed:   (2^0)th term = 1st term =  (3^0) (2^1)th term = 2nd term = (3^1) (2^2) term = 4th term = (3^2) similarly (2^6)th term = 64th term = (3^6) = 729 add (2^5)th term to the above = 729 + 243 add 4th term to the above = 729 + 243 + 9

regards,
thecuriousone
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Aryabhatta
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 Re: 100th number in sequence   « Reply #4 on: Jun 16th, 2007, 8:23pm » Quote Modify

You got the right answer, thecuriousone.

Welcome to the forums!
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thecuriousone
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 Re: 100th number in sequence   « Reply #5 on: Jun 17th, 2007, 2:15am » Quote Modify

Thanks, Aryabhatta!

But I am sure, you have a more elegant solution than what I wrote.
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towr
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 Re: 100th number in sequence   « Reply #6 on: Jun 17th, 2007, 7:06am » Quote Modify

on Jun 17th, 2007, 2:15am, thecuriousone wrote:
 But I am sure, you have a more elegant solution than what I wrote.
I'm pretty sure that what he had in mind is basicly the same.

Write 100 as binary, 100=64+32+4 = 26+25+22 = 11001002 (the subscript 2 is to denote it is in base 2, or binary). Next interpret this numberstring as a number in base 3, 11001003 = 36+35+32 = 729 + 243 + 9 = 981.

This is essentially what you're doing. Even though you don't explicitly resort to binary.
Recognizing the intermediate step as binary makes it a bit simpler if you're well-versed in dealing with other number bases. Which is why Grimbal wanted to write the answer in base 3, then you don't need the last step converting it to decimal
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Wikipedia, Google, Mathworld, Integer sequence DB
Aryabhatta
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 Re: 100th number in sequence   « Reply #7 on: Jun 19th, 2007, 1:44am » Quote Modify

towr is right. The basic idea is the same, the presentation differs.
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