find 4 numbers forming a GP in which the 3rd term is >1st by 9 & the2nd term is > the 4th by 18.?

Question

Please answer with steps.

Answer 1

A geometric progression.

Let's call the first term x. Then the third term is x + 9.

Let's call the fourth term y. Then the second term is y + 18.

The four terms of the progression are

x, y + 18, x + 9, y

and adjacent terms have a common factor.

(1)... (y + 18) / x = common factor

and

(2)... (x + 9) / (y + 18) = common factor

and

(3)... y / (x + 9) = common factor

We have three equations and three unknowns. Let's call the common factor f.

From (1)

(4)... x = (y + 18) / f

Inserting that into (3), we get

(5)... y / ((y + 18) / f + 9) = f
(6)... y = f * ((y + 18) / f + 9)
(7)... y = (fy + 18f) / f + 9f
(8)... fy = fy + 18f + 9f^2
(9)... f^2 + 2f = 0
(10)... f * (f + 2) = 0
(11)... f = -2

Now we have to find either x or y.

It can be done algebraically, but a little trial and error shows that x = 3, and the four terms are

3, -6, 12, -24