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   100th number in sequence
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   Author  Topic: 100th number in sequence  (Read 4744 times)
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?  Roll Eyes
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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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Smiley
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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?  Cool
 
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
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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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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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