Three positive integers are chosen at random without replacement from 1,2,....,64.
What is the probability that the numbers chosen are in geometric sequence?
Order of choice doesn't matter. For example, 412 would qualify as
numbers in geometric sequence.
Bonus Question:
Generalise this result (in terms of n) covering the situation where three positive integers are chosen at random without replacement from 1,2,.....,n^{2}.
I now count 58 geometric sequences. Each of the following is a possible first term followed the ratio of subsequent terms.
Squarefree numbers must have integer ratios but numbers that are not squarefree can have noninteger ratios and still have integer terms.
1 ratios 2, 3, 4, 5, 6, 7, 8
2 ratios 2, 3, 4, 5
3 ratios 2, 3, 4
4 ratios 3/2, 2, 5/2, 3, 7/2, 4
5 ratios 2, 3
6 ratios 2, 3
7 ratios 2, 3
8 ratios 3/2, 2, 5/2
9 ratios 4/3, 5/3, 2, 7/3, 8/3
10 ratio 2
11 ratio 2
12 ratios 3/2, 2
13 ratio 2
14 ratio 2
15 ratio 2
16 ratios 5/4, 6/4, 7/4, 2
18 ratios 4/3, 5/3
20 ratio 3/2
24 ratio 3/2
25 ratios 6/5, 7/5, 8/5
27 ratio 4/3
28 ratio 3/2
32 ratio 5/4
36 ratios 7/6, 8/6
49 ratio 8/7
This makes a general count leading to the bonus much more complicated than before.
Edited on February 28, 2013, 1:42 pm

Posted by Jer
on 20130228 11:30:27 