|
||
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:
[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. |
||
Powered by YaBB 1 Gold - SP 1.4! Forum software copyright © 2000-2004 Yet another Bulletin Board |