Sequence Pattern and Closed Form
Date: 7/16/96 at 3:19:18 From: Anonymous Subject: Sequence Pattern and Closed Form No. of Discs 1 2 3 4 5 6 7 8 9 10 No. of moves 1 3 7 15 31 63 127 255 511 1023 ^ ^ ^ ^ ^ ^ ^ ^ ^ 2 4 8 16 32 64 128 256 512 I had to continue the pattern after being told the first 4 numbers of discs and moves. I figured out the rest by doubling the difference between the moves bit and then realized that M is the difference minus one. I cannot figure out the rule for the table though. I have spent ages on it but I can't do it. Could you help? (Please? :) Thanks
Date: 7/16/96 at 11:24:13 From: Doctor Jodi Subject: Re: Sequence Pattern and Closed Form Hi there! The number of moves, expressed in relation to the number of discs is (2^(number of discs) )- 1 That is, 2 to the power of the number of discs minus 1. Hope this helps. Let us know if you have any other questions. -Doctor Jodi, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 7/16/96 at 15:30:47 From: Doctor Anthony Subject: Re: Sequence Pattern and Closed Form If you had continued the difference table to another row, you would have found that you repeated the numbers 2, 4, 8, 16, 32, ... When you have this situation you can be sure that if n is the number of the term, then n will be a power in the expression for the nth term. By inspection the terms are generated by the function 2^n - 1 n 2^n -1 ----------- 1 1 2 3 3 7 4 15 5 31 - - and so on as we can see this function is generating the terms of the sequence. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.