Find the probability that $3$ distinct numbers chosen from $\{1,2,\ldots,n\}$ form a geometric progression. 
Find the probability that $3$ distinct numbers chosen from $\{1,2,\ldots,50\}$ form a geometric progression.

Choose numbers $a,ar,ar^2$ such that $1\le a,ar,ar^2\le 50$.
I want to know if there's a straightforward (less casework) way to go about the case where the common ratio is not an integer.
Here's my work for your reference:
Choose $a, ar,ar^2$. Make cases for $r$, and deduce the permissible range for $a$. Note that $r\le 7$. When $r\in\mathbb N$:
$$\frac{12 + 5 + 3 + 2 + 1 + 1}{{50\choose 3}}  = \frac{24}{{50\choose 3}}$$
When $r\in\mathbb{Q}\setminus\mathbb N$, choose $r = m/n$ where $m,n$ are co-prime and $m>n>1$. Note that $1 \le a\le ar \le ar^2 \le 50$ so $$1 \le a\le \left\lfloor \frac{50n^2}{m^2}\right\rfloor$$
where $n^2|a$. The probability in this case is:
$$\frac{20}{{50\choose 3}}$$
So the probability is $\displaystyle \frac{44}{{50\choose 3}}$.
Edit:So there's a nice closed form in the comments, how is it derived?
 A: As ACheca points out, there is a closed-form formula for the number 3-term geometric progressions in $\{1,\ldots,n\}$:
$$\sum_{1<m^2\leq n}\phi(m)\lfloor n/m^2\rfloor.$$
Here is an explanation. The number of 3-term GPs is equal to the number of solutions to $ab=c^2$ with $a,b,c\in \{1,\ldots,n\}$ and $a<b$. (These give exactly 3-term GPs, with $b=ax^2$ and $c=ax$.)
If we let $d=(a,b)$, then both $b/d$ and $a/d$ must be squares, say $m^2$ and $k^2$ respectively. Note that since $a<b$ we have $1\leq k<m$.
Thus the number of 3-term GPs is equal to the number of $1\leq d\leq n$ and $1< m^2\leq n$ and $1\leq k<m$ such that $(k,m)=1$ and $1\leq m^2d\leq n$. (Since any such choice uniquely determines a valid $a,b$ and hence $c$, and vice versa.)
Thus the total count is equal to
$$\sum_{1\leq d\leq n}\sum_{1<m^2\leq n}\left(\sum_{1\leq k<m}1_{(k,m)=1}\right)1_{1\leq m^2d\leq n} = \sum_{1\leq d\leq n}\sum_{1<m^2\leq n}\phi(m)1_{1\leq m^2d\leq n}=\sum_{1<m^2\leq n}\phi(m)\lfloor n/m^2\rfloor $$
as required.
