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Topic: 100th number in sequence (Read 4879 times) |
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Aryabhatta
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100th number in sequence
« on: Jun 16th, 2007, 9:40am » |
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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 » |
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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 » |
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thecuriousone
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Re: 100th number in sequence
« Reply #3 on: Jun 16th, 2007, 12:02pm » |
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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 » |
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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 » |
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Thanks, Aryabhatta! But I am sure, you have a more elegant solution than what I wrote.
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towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
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Re: 100th number in sequence
« Reply #6 on: Jun 17th, 2007, 7:06am » |
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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
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Aryabhatta
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Re: 100th number in sequence
« Reply #7 on: Jun 19th, 2007, 1:44am » |
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towr is right. The basic idea is the same, the presentation differs.
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