Question

Three nonzero real numbers form an arithmetic progression and the squares of those numbers taken in the same order form a geometric progression. Find all possible common ratios of the geometric progression.

Answer 1

Let the numbers be $a-d,a,a+d$

Squares of these number form a GP

$(a-d{)}^2,a^2,(a+d{)}^2$

$\Rightarrow (a^2{)}^2=(a-d{)}^2(a+d{)}^2$

$\Rightarrow a^4=(a^2-d^2{)}^2$

$\Rightarrow a^2=a^2-d^{2}or{a}^2=d^2-a^2$

$\Rightarrow d=0ord=\pm \sqrt{2}a$

From 1st case not possible as the numbers are distinct

$r=\frac{a^2}{(a-d{)}^2}$

also $r=\frac{(a+d{)}^2}{a^2}$

substitue d as obtained above and get the possible values of r .