Question

Let $a < b < c $ be three integers such that $a, b, c $ is an arithmetic progression and $a, c, b$ is a geometric progression, then find the smallest positive value of c

Answer 1

$(A+d)^2=(A-d) A $
$A^2+d^2+2 A d=A^2-A d $
$d^2=-3 A d $
$d=0, d=-3 A $
$c=A+d=-2 A$
$\therefore c_{\min }=2 .$