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All rational numbers in (Posted on 2016-08-26) Difficulty: 3 of 5
The infinite sequence is defined as follows:
for n ≥ 1, a2n= an+1
a2n+1=1/ a2n

Prove that every positive rational number occurs in the sequence exactly once.

No Solution Yet Submitted by Ady TZIDON    
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From N/D to n | Comment 3 of 10 |
How to find the position of a given rational number.  Create a recursive sequence beginning with the number and ending with 1 using the following rule:
If the number is less than 1, create its reciprocal;
if the number is greater than 1, subtract 1;
stop when 1 is reached.
(keep track of the order of these steps.)

Example beginning with 7/16 (step in parentheses) last column explained below
7/16 (r) 101
16/7 (-1) 100
9/7 (-1) 50
2/7 (r) 25
7/2 (-1) 24
5/2 (-1) 12
3/2 (-1) 6
1/2 (r) 3
2 (-1) 2

The last column is the value of n for each value in the first column and is found by starting with 1 at the bottom and working up by the following rule:
If there is a (-1) then multiply by 2;
if there is a (r) then add 1.

So the number a101=7/16.

It seems clear that each rational number should reach 1 in a unique way and that the reversal process should reach a unique value.  I have not actually proven this.

Edit was to fix a typo.  I made a different change than the one Charlie suggested.

Edited on August 29, 2016, 8:42 am
  Posted by Jer on 2016-08-28 17:09:33

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