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Topic: 100th number in sequence (Read 4874 times) 

Aryabhatta
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100th number in sequence
« on: Jun 16^{th}, 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 100^{th} number of this sequence.


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Grimbal
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Re: 100th number in sequence
« Reply #1 on: Jun 16^{th}, 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 16^{th}, 2007, 10:19am » 
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thecuriousone
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Re: 100th number in sequence
« Reply #3 on: Jun 16^{th}, 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 16^{th}, 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 17^{th}, 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
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Re: 100th number in sequence
« Reply #6 on: Jun 17^{th}, 2007, 7:06am » 
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on Jun 17^{th}, 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 = 2^{6}+2^{5}+2^{2} = 1100100_{2} (the subscript 2 is to denote it is in base 2, or binary). Next interpret this numberstring as a number in base 3, 1100100_{3} = 3^{6}+3^{5}+3^{2} = 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 wellversed 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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Aryabhatta
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Re: 100th number in sequence
« Reply #7 on: Jun 19^{th}, 2007, 1:44am » 
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towr is right. The basic idea is the same, the presentation differs.


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