If three non-zero distinct real numbers form an arithmetic progression and the squares of these numbers taken in the same order constitute a geometric progression, then find the sum of all possible common ratios of the geometric progression.

Hi Student

Let the numbers be a-d,a,a+d
Squares of these number form a GP
(a-d)² , a² , (a+d)²

=> (a²)² = (a-d)²(a+d)²
= > a⁴ = ( a²-d²)²
=> a² = a²-d² or a²= d²-a²
=> d=0 or d= ± a
From 1^st case not possible as the numbers are distinct

r= a²./ (a-d)² also r = (a+d)²/a²
substitue d as obtained above and get the possible values of r .