If the i-th, j-th, and k-th terms in an arithmetic progression are in a geometric progression with ratio r, find r in terms of i, j, and k.
This is my result:
(1) if $ik \ne j^2$ then $r=\frac{k-j}{j-i}$; (2) if $ik = j^2$ then $r=\frac{j}{i}$.
I solved this here (Arithmetic and Geometric Progression Question 1) but I feel my proof is awkward, and my hope is that someone can come up with a more elegant solution.