Two positive numbers (a,b). One G.M. (G=\sqrt{ab}).

Two A.Ms (p,q): sequence (a,p,q,b) [ p=a+d,\ q=a+2d,\ b=a+3d ] [ d=\dfrac{b-a}{3},\quad p=\dfrac{2a+b}{3},\quad q=\dfrac{a+2b}{3} ]

[ 2p-q=\dfrac{4a+2b-a-2b}{3}=a,\quad 2q-p=\dfrac{2a+4b-2a-b}{3}=b ]

[ (2p-q)(2q-p)=ab=G^2 ]

Proved.