The question you opened
Nine, three, one, a third
Each number in this list is smaller than the one before: 9, 3, 1, one third.
What number comes next?
1/9
Each term is the one before divided by 3. A third divided by 3 is a ninth, and the sequence never reaches zero.
The sequence runs 9, 3, 1, one third, and the next term is one ninth. Differences do not help here. From 9 to 3 the drop is 6, then 2, then two thirds. Those gaps are shrinking in a way that is hard to read directly. Ratios do help, and ratios are the second thing to try whenever differences look untidy.
Divide each term by the one before it: 3 over 9 is a third, 1 over 3 is a third, a third over 1 is a third. The ratio is constant, so this is a geometric sequence with a common ratio of one third. Apply the ratio once more. One third divided by three is one ninth. Keep going and you get one twenty seventh, one eighty first, and so on.
The interesting property of this sequence is that it never arrives at zero. Each term is a third of the previous one, so it is always positive and always smaller. It approaches zero without ever landing there, which is why zero is a tempting answer and a wrong one. Sequences that shrink by a fixed proportion behave very differently from sequences that shrink by a fixed amount.
Subtracting 3 each time from 9 reaches zero at the fourth step. Dividing by 3 each time never does. That distinction is worth more than the puzzle. Anything that decays proportionally, a medicine leaving the bloodstream, a sound fading, a debt paid down by a fixed percentage, follows the same shape: fast at first, then a long tail that never quite ends.
The practical checklist for any number sequence is short. Take the differences. If they are constant, you have an arithmetic sequence. If not, take the ratios. If those are constant, you have a geometric one. If neither works, look at the differences of the differences. Three checks, in that order, will crack most sequences you meet.
Technique: Test the ratio on every pair