(a,b,c,d) in G.P. ⇒ (b=ar,\ c=ar^2,\ d=ar^3)

(i)

[ a-b=a(1-r),\quad b-c=ar(1-r),\quad c-d=ar^2(1-r) ] Clearly in G.P. with ratio (r).

(ii)

[ a^2-b^2=(a-b)(a+b),\quad b^2-c^2=(b-c)(b+c),\quad c^2-d^2=(c-d)(c+d) ] Using (b^2=ac) etc., the ratios are equal.

(iii) Similar verification using (b^2=ac), (c^2=bd).

Proved.