<s>[INST] Proof of rational numbers I am recently starting in the subject of pure mathematics and this problem has eluded me. 
Any rational number $\frac{p}{q}$ can be expressed in a simple series form:
$$\frac{1}{1},\frac{2}{1},\frac{1}{2},\frac{3}{1},\frac{2}{2},\frac{1}{3},\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4},...$$
Show that $\frac{p}{q}$ is the $[\frac{1}{2}(p+q-1)(p+q-2)+q]th$ of the series (where $p$ and $q$ are coprime and $p \neq 0$).
 [/INST] Hint. Divide the numbers in rows:
$$\frac{1}{1}$$
$$\frac{2}{1},\frac{1}{2}$$
$$\frac{3}{1},\frac{2}{2},\frac{1}{3}$$
$$\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4}.$$
In the first $n$ rows there are $1+2+\dots+n=n(n+1)/2$ fractions.
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