The real positive numbers a,b,c,d are the terms of a geometric progression. What is the common ratio if d-a=7 and c-b=2?

We have four terms a, b, c, d in geometric progression
with d - a = 7 and c - b = 2. We have to find the common
ratio.

Let the common ratio be r, this gives b = ar, c =
ar^2 and d = ar^3

ar^3 - a = 7 and ar^2 - ar =
2

ar^2 - ar = 2

=> a =
2/(r^2 - r)

substitute in ar^3 - a =
7

=> (r^3 - 1)*2/(r^2 - r) =
7

=> 2*r^3 - 2 = 7r^2 -
7r

=> 2r^3 - 7r^2 + 7r - 2 =
0

=> 2r^3 - 4r^2 - 3r^2 + 6r + r - 2 =
0

=> 2r^2(r - 2) - 3r(r - 2) + 1(r - 2) =
0

=> (2r^2 - 3r +1 )(r - 2) =
0

=> (2r^2 - 2r - r + 1)(r - 2) =
0

=> (2r(r - 1) - 1(r - 1))(r - 2) =
0

=> (2r - 1)(r - 1)(r - 2) =
0

=> r = 0.5 and r = 1 and r =
2

The possible values of the common ratio are
(0.5, 1, 2)