@Placholdr
The solution I would give goes something like this:
The sequence is made up of smaller subsequences (e.g. "3/1, 2/2, 1/3") and the k-th subsequence has exactly k terms.

We know p/q is the q-th term in the (p+q)-th subsequence. Before the (p+q)-th subsequence there have been p+q-1 subsequences, and the total length of those subsequences is 1 + 2 + 3 + ... + (p+q-1).

We can use the formula for the triangle numbers: 1 + 2 + 3 + ... + m = 1/2m(m-1).
Filling in (p+q-1) for m there gives us the number of terms before the (p+q)-th subseguence starts, and adding q gets us to the right term within the (p+q)-th subsequence, and with that we get exactly the formula for n that is given.