Suppose you have a collection of large amount of Hydrogen atoms in $n$th state($n-1$th excited state). They have to go to their ground state($n$=1).
Going from $n_1$ to $n_2$ makes a unique spectral line. An atom cannot raise its $n$, it can only decrease it.What is the minimum number of $H$ atoms needed to view all spectral lines? Although changing of $n$ is random, we may assume that we are lucky.e.g. for $n=3$ :
We have to see $$3 \to 1,2\to 1, 3\to 2$$ Minimum number of atoms is $2$. One will go to $3 \to 2 \to 1$ and other will $3\to 1$.
We have to see $4\to 3,3\to 2,2\to 1,4\to 1, 4\to 2,3\to 1$ Minimum number is $4$ :
$$4\to 1, 4 \to 3 \to 1,4\to 2\to 1,4\to3\to2\to1$$
How can we generalize it for any $n$? Please add appropriate tags.