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riddles >> easy >> 100th number in sequence
(Message started by: Aryabhatta on Jun 16th, 2007, 9:40am)

Title: 100th number in sequence
Post by Aryabhatta on Jun 16th, 2007, 9:40am
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.

Title: Re: 100th number in sequence
Post by Grimbal on Jun 16th, 2007, 9:56am
[hide]Can I write the number in base 3, for added difficulty?[/hide]  ::)

Title: Re: 100th number in sequence
Post by Aryabhatta on Jun 16th, 2007, 10:19am
:-)

Title: Re: 100th number in sequence
Post by thecuriousone on Jun 16th, 2007, 12:02pm
[hideb]is it 981?  8)

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[/hideb]

regards,
thecuriousone

Title: Re: 100th number in sequence
Post by Aryabhatta on Jun 16th, 2007, 8:23pm
You got the right answer, thecuriousone.

Welcome to the forums!

Title: Re: 100th number in sequence
Post by thecuriousone on Jun 17th, 2007, 2:15am
Thanks, Aryabhatta!

But I am sure, you have a more elegant solution than what I wrote.

Title: Re: 100th number in sequence
Post by towr on Jun 17th, 2007, 7:06am

on 06/17/07 at 02:15:05, 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.

[hide]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[/hide]

Title: Re: 100th number in sequence
Post by Aryabhatta on Jun 19th, 2007, 1:44am
towr is right. The basic idea is the same, the presentation differs.



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