Find the next three in the sequence: 0.3333..., 0.3, 0.252525..., 0.222..., 0.2, 0.131313..., ?, ?, ?

This is probably wrong, but it’s my attempt at the answer.

If you write out the numbers in fractional form, they are 3/9, 3/10, 25/99, 2/9, 2/10, 13/99.

Well, notice the denominators go 9, 10, 99, 9, 10, 99. So I think the next three numbers will follow the same pattern and be a/9, b/10, c/99.

Now for the numerators. Well, I broke this up and looked at it like 3 intertwined sequences. I other words, I saw it as
3/9, 2/9, a/9
3/10, 2/10, b/10
25/99, 13/99, c/99

Well, it seems like a and b are probably 1 since the numbers are decreasing.
I couldn’t find an obvious relationship between 25 and 13. The best I could do is notice that 13 = 25/2 rounded up. If that is the case, it fits for 2 = 3/2 rounded up and 1 = 2/2 rounded up.

So that would make the next three numbers 1/9, 1/10, 7/99, or
0.111…, 0.1, 0.070707…

Another solution could be that the relationship 13 = (25+1)/2. This would still make c = (13+1)/2 = 7, but a and b change. a = b = (2+1)/2 = 1.5. In that case the next three numbers are 1.5/9, 1.5/10, 7/99, or
0.1666…, 0.15, 0.070707…

Posted by nikki on 2004-11-30 13:32:06